On the diffusive structure for the damped wave equation with variable coefficients. Yuta Wakasugi

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1 On the diffusive structure for the damped wave equation with variable coefficients Yuta Wakasugi Department of Mathematics, Graduate School of Science, Osaka University Doctoral thesis 4

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3 Contents Chapter. Introduction 5.. Damped wave equations 5.. Results 8.3. A review of some previous results.4. Methods for proof 38 Chapter. Basic facts and their proof 43.. Linear damped wave equations 43.. Diffusion phenomenon 5.3. Semilinear damped wave equations with a source nonlinearity 6 Chapter 3. On diffusion phenomena for the linear wave equation with space-dependent damping Introduction and results Basic weighted energy estimates Weighted energy estimates for higher order derivatives Proof of the main theorem 4 Chapter 4. Small data global existence for the semilinear wave equation with damping depending on time and space variables Introduction and results A priori estimate 4.3. Estimates of the lifespan from below 7 Chapter 5. Critical exponent for the semilinear wave equation with scale-invariant damping Introduction and results Proof of global existence Proof of blow-up 38 Chapter 6. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables Introduction and results Existence of multiplier function Proof of blow-up The case of perturbation of +t Proof of Lemma Proof of Theorems 6.7 and

4 4 CONTENTS Chapter 7. Estimates of the lifespan Effective damping cases Proof of the estimates of the lifespan Scale-invariant damping case Time and space dependent damping case 66 Chapter 8. A remark on L p -L q estimates for solutions to the linear damped wave equation Introduction Proof of Lemma Proof of Theorem 8. 7 Chapter 9. Appendix Notation Special functions Lemmas Definition of solutions Local existence 3 Bibliography 7 Acknowledgements 3 List of the author s papers cited in this thesis 5

5 CHAPTER Introduction The damped wave equation.. Damped wave equations u tt u + u t = is a model which describes the propagation of the wave with friction. This equation is also known as the telegraph equation developed by Oliver Heaviside It has been investigated by many mathematicians that the solution of the damped wave equation has so-called diffusion phenomenon, that is, the solution behaves like that of the corresponding heat equation v t v = as t +. In this thesis we are concerned with the damped wave equation with variable coefficients u tt u + at, xu t = fu. Our aim is to study whether this equation still has a diffusive structure. Here at, x denotes a coefficient of the damping depending on time and space variables and fu is a nonlinear term. Roughly speaking, it is expected that if at, x does not decay fast, then the damping is effective and the solution behaves like that of the corresponding heat equation v + at, xv t = fv; if at, x decays sufficiently fast, then the damping becomes non-effective and the solution behaves like that of the wave equation without damping w tt w = fw. There are many literature on the damped wave equation with variable coefficients and the above conjecture has been confirmed in several situations. In this thesis, we focus on the damping of the form at, x = + x α/ + t β with α, β. In this case, roughly speaking, it is known that if α + β <, then at, x is effective damping and if α + β >, then at, x is non-effective damping. Here we give an intuitive observation for understanding the diffusion phenomenon. Consider the linear damped wave equation.. u tt u + u t = with initial data u, u t, x =, gx. Using the Fourier transform, we have û tt + ξ û + û t =. 5

6 6. INTRODUCTION Solving this ordinary differential equation, one can obtain ût, ξ = e t 4 ξ / + e t+ 4 ξ / ĝξ. 4 ξ When ξ is sufficiently small, we can consider 4 ξ ξ = + 4 ξ ξ and hence, if t is sufficiently large, ignoring some terms decaying exponentially, we can see that ût, ξ e t ξ ĝξ. The right-hand side is of course the Fourier transform of the solution of the corresponding heat equation with initial data g. Thus, we can naturally expect the diffusion phenomenon. We also give an another observation by scaling argument. For a solution ut, x of.., putting ut, x = ϕλt, λ / x, with a parameter λ >, we have λt = s, λ / x = y λϕ ss s, y ϕs, y + ϕ s s, y =. Thus, letting λ, we obtain the heat equation ϕ + ϕ s =. We note that λ is corresponding to t +. This observation is also applicable to variable coefficient cases. Let ut, x be a solution of u tt u + x α t β u t =. When < β <, α < and α + β <, we put ut, x = ϕλ /+β t, λ / α x, with a parameter λ > and have λ /+β t = s, λ / α x = y λ /+β ϕ ss s, y λ / α ϕs, y + λ α/ α+ y α s β ϕ s s, y =. We can rewrite this equation as λ /+β / α ϕ ss ϕ + y α s β ϕ s =. Note that + β α β = α + β α >. Therefore, letting λ again, we obtain the corresponding heat equation ϕ + y α s β ϕ s =. On the other hand, when α + β >, we put ut, x = ϕλt, λx, λt = s, λx = y and have ϕ ss s, y ϕs, y + λ α+β y α s β ϕ s s, y =. In this case, letting λ, we obtain the wave equation without damping ϕ ss ϕ =.

7 .. DAMPED WAVE EQUATIONS 7 This indicates that the asymptotic behavior of solutions essentially depends on the behavior of the coefficient of damping. In this thesis, we investigate in what way the damping influences the behavior of solutions. This thesis is organized as follows. In the next section, we introduce selected results described in the following chapters. Then, in Section.3, we give a review of previous results related to ours. We also explain the method used for the proof of main results in Section.4. In Chapter, we introduce some basic results on the study of damped wave equations, including solution representation formula, asymptotic behavior of solutions and some semilinear problems. Chapter 3 concerns with the diffusion phenomena for the linear wave equation with space-dependent damping. We prove the asymptotic profile of the solution is given by a solution of the corresponding heat equation in the L -sense. We also give weighted energy estimates of solutions for higher order derivatives. Chapter 4 is devoted to the existence of global solutions for the semilinear wave equation with damping depending on time and space variables. In this case we can find an appropriate weight function related to the corresponding heat kernel and we can obtain an a priori estimate of the solution by a weighted energy method. As a corollary of this a priori estimate, we can see that the energy of the solution is concentrated in some parabolic region much smaller than the light cone. This behavior is quite a contrast to that of the wave equation without damping. In Chapter 5, we consider the critical exponent problem to the semilinear wave equation with scale-invariant damping µ +t u t with µ >. This equation is invariant under the hyperbolic scaling and known as the threshold between effective and non-effective damping. In this case the asymptotic behavior of the solution is very delicate and the coefficient µ plays an essential role. We prove an L L type decay estimate of solutions and a small data global existence result for sufficiently large µ. We also show some blow-up results for all µ > by using a modified test function method. Moreover, we prove that when µ <, the critical exponent is larger than that of the corresponding heat equation. This shows that the equation loses the parabolic structure and recovers its hyperbolic structure as µ gets smaller. Chapter 6 concerns with the blow-up of solutions to the one-dimensional semilinear wave equation with time and space variables. In this case we cannot apply the test function method directly. However, in one-dimensional case, we can construct an appropriately multiplier function by the method of characteristics. We prove that when the damping is non-effective, the critical exponent agrees with that of the wave equation without damping, that is, the small data blow-up holds for any power of the nonlinearity. In Chapter 7, we prove upper estimates of the lifespan of solutions to the semilinear wave equation with several types of damping in subcritical case. In particular, our results give almost optimal estimates of the lifespan from both above and below in the constant and time-dependent coefficient cases. This is a joint work with Mr. Masahiro Ikeda. In Chapter 8, we prove the L p -L q estimates of the solution to the linear damped wave equation. These estimates, which show the diffusion phenomenon, has been already proved by several mathematicians. In this chapter we introduce an improvement of these estimates in higher dimensional cases and give a simpler proof

8 8. INTRODUCTION by using the solution representation formula. This is a joint work with Mr. Shigehiro Sakata. Finally, in Appendix, we explain the notation used in this thesis, some useful lemmas, definitions of solutions and the proof of local existence theorem... Results In this section, we collect the selected results described in this thesis. For the sake of simplicity, we introduce only simplified results. We state the results more precisely in the following chapters. We consider the Cauchy problem of the semilinear damped wave equation { utt u + at, xu.. t = fu, t, x, R n, u, u t, x = u, u x, x R n, where u = ut, x is real-valued unknown, at, x is nonnegative smooth function and fu = or u p. In what follows, we assume that < p and u, u C R n C R n. The first result is about the asymptotic profile of solutions to the linear wave equation with the damping having the form x α with α <. In this case the damping can be seen as effective and it is conjectured the asymptotic profile is given by a solution of the corresponding heat equation. The following result gives an affirmative answer and is explained in Chapter 3: Theorem. []. Let fu = and at, x = x α with α < and let u be the solution of... Then we have ut, vt, L = ot n α α as t +, where vt, x is the solution of the corresponding heat equation x α v t v = with the initial data v, x = u x + ax u x. The second result is the existence of global solutions to the semilinear wave equation with damping depending on time and space variables. We consider the damping of the form x α + t β with α, β, α + β < and the nonlinearity u p. We prove that if p > + /n α, then the global solution exists for small data having the finite weighted energy. We note that the exponent + /n α agrees with the critical exponent of the corresponding heat equation. Theorem. [9]. Let fu = u p and at, x = a x α + t β with a >, α, β, α + β <, and let ψt, x = A x α + t +β, A = + βa α + δ with δ >. If + n α < p n n 3, + < p < n =,, n n α then there exists a small positive number δ > such that for any < δ δ the following holds: If I := e ψ,x u x + u x + u x dx R n

9 .. RESULTS 9 is sufficiently small, then there exists a unique solution u C[, ; H R n C [, ; L R n to.. satisfying where R n e ψt,x ut, x dx C δ I + t +β R n e ψt,x u t t, x + ut, x dx C δ I + t +β ε = εδ := and C δ is a constant depending on δ. 3 + βn α α + δ δ n α α +ε, n α α ++ε, The following two theorems are about the critical exponent problem for the semilinear wave equation with the scale-invariant damping a = µ/+t. It is known that if µ is sufficiently large resp. small, the solution of the equation with fu = behaves like the corresponding heat equation resp. the free wave equation. We give a global existence result for p > + /n, provided that µ is sufficiently large. The exponent + /n is known as critical for the corresponding heat equation. Thus, in view of the linear problem, the assumption on µ is reasonable. We also give a blow-up result for all µ >. We remark that if µ <, then the blow-up result holds even when p > + /n. This phenomena can be interpreted as that if µ is small, then the equation recover the hyperbolic structure and the critical exponent rises. Theorem.3 []. Let fu = u p and at, x = µ +t ψt, x = A x + t, A = µ + δ. with µ >, and let If + n < p n/n n 3, + n < p < n =, and < ε < np + n /p, then there exist constants δ > and µ > having the following property: if µ µ and I = e ψ,x u x + u x + u x dx R n is sufficiently small, then there exists a unique solution u C[, ; H R n C [, ; L R n of.. satisfying R n e ψt,x ut, x dx C µ,ε I + t n+ε, R n e ψt,x u t t, x + ut, x dx C µ,ε I + t n +ε for t, where C µ,ε is a positive constant depending on µ and ε. Theorem.4 []. Let fu = u p and at, x = µ +t, µ >. i < p + /n and µ >. Moreover, we assume that lim inf µ u + u dx >. R x <R Then there is no global solution for...

10 . INTRODUCTION iilet < µ and We also assume < p + n + µ. lim inf u xdx >. R x <R Then there is no global solution for... The next one is a blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We consider a noneffective damping and prove a blow-up result for any < p <. This is corresponds to the result of the corresponding semilinear wave equation without damping. Theorem.5 []. Let n = and fu = u p, and assume that at, x C [, R satisfies α t β x at, x δ + t k+α α, β =, with some k > and sufficiently small δ >. If < p < and u =, u, lim inf u xdx >, R then there is no global solution... x <R The following result is about estimates of the lifespan of solutions to the semilinear wave equation with time or space dependent damping. Even for the constant damping case, to obtain the estimates of lifespan was open problem for higher dimensional case n 4. Here we give the optimal estimate from above for the constant and time-dependent damping case. We also give an upper estimate for space-dependent damping, which does not seem to be optimal, but the first result for this area. Theorem.6 [6]. Consider the initial data εu, u instead of u, u in.., where ε > is small parameter. Let fu = u p and at, x = x α + t β with α [,, β,, αβ =, and let < p < + /n α. We assume that the initial data u, u satisfy lim inf x α Bu x + u xdx >, R where x <R B = e t +s β ds dt. Then there exists C > depending only on n, p, α, β and u, u such that the lifespan T ε is estimated as ε /κ if + α/n α < p < + /n α, T ε C ε p logε p if α >, p = + α/n α, ε p if α >, < p < + α/n α

11 .3. A REVIEW OF SOME PREVIOUS RESULTS for any ε, ], where κ = + β α p n α. The final result is the L p -L q estimate of the solution to the linear damped wave equation. This has already shown by Marcati and Nishihara [58], Hosono and Ogawa [3] and Nishihara [78] for the case n =,, 3, respectively. Narazaki [73] proved the same type estimate with arbitrary small loss of decay rate ε. In this thesis we prove that the loss ε can be removed and give a simpler proof by using the solution representation formula. Let u, v be the solutions to the linear damped wave equation and the corresponding heat equation { utt u + u t =, t, x, R n, u, u t, x =, gx, x R n, respectively. { vt v =, t, x, R n, v, x = gx, x R n, where Theorem.7 [98]. For q p and t >, we have if n =, W n tg = ut vt e t/ W n tg L p Ct n q p g L q, W tgx = n 3/ n!! S n 8 l l! t l= if n 3 and an odd number, x+t x t t gsds n / W n tg = n!! S n 8 l l! t t l= t x y gydy x y t n 3/ l gyds y t x y =t n / l if n is an even number, where S n denotes the measure of the n-dimensional unit sphere..3. A review of some previous results In this section, we give a review of previous study for the damped wave equation with variable coefficients from the view point of the diffusion phenomenon. We clarify the relation between the results stated in the previous section and some earlier literature.

12 . INTRODUCTION.3.. Linear damped wave equations. We consider the linear damped wave equation { utt u + at, xu.3. t =, t, x, R n, u, u t, x = u, u x, x R n, where u = ut, x is a real-valued unknown function and u, u is given initial data. The simplest case is the constant coefficient case, that is { utt u + u.3. t =, u, u t, x = u, u x. As we mentioned in the first section, it is well-known that the solution of.3. has the diffusion phenomenon. This means that the solution of.3. behaves like the solution of the corresponding heat equation { vt v =,.3.3 v, x = u x + u x as t +. First, we note that by the Duhamel principle, the solution u of.3. is expressed by ut, x = S n tu + u + t S n tu, where S n t denotes the solution operator of the Cauchy problem { utt u + u t = t, x R R n, u, u t, x =, gx x R n, that is, S n tg is the solution to the above problem. The asymptotic behavior of solutions to the equation.3. has been initiated by Matsumura [59]. He proved the following estimates by using the Fourier transform:.3.4 t i x α S n tg L C + t n/m i α / g L m + g H [n/]+i+ α, t i x α S n tg L C + t n/4 n/m i α / g L m + g H i+ α, where m. The decay rates above are the same as those of the corresponding heat equation.3.3. The study of the precise asymptotic profile of solutions to.3. was triggered by the observation by Hsiao and Liu [4]. They considered a system of hyperbolic conservation laws with damping.3.5 V t U x =, U t + pv x = αu, V, U, x = V, U x V ±, U ±, x ±, where t, x, R, α >, pv >, p V < for V > and V, V ± >. They proved that the asymptotic profile of the solution V, U of.3.5 is given by a solution of a system given by Darcy s law: { Vt = α p V xx,.3.6 p V x = αū, i.e., { Vt Ūx =, p V x = αū with V, ± = V ±. By putting W t, x = x V t, y V t, y +x ˆV t, ydy with some auxiliary function ˆV and suitably chosen x, they reduced the system

13 .3. A REVIEW OF SOME PREVIOUS RESULTS to a quasilinear second order hyperbolic equation with damping.3.7 { Wtt + αw t + pw x + V + ˆV pw x x = α p V, W, W t, x = W, W x as x ±. Using this formalization, they obtained V V t L L = Ot /, t +, here f L L := max{ f L, f L}. Under the additional assumption V, U ± = V,, V x V dx = with some V >, this convergence rate was improved by Nishihara [76] to V V t L = Ot. Moreover, he also consider in [77] a generalization of.3.7 in one space dimension { utt + αu.3.8 t au x x =, u, u t, x = u, u x and proved that the solution of.3.8 behaves like that of the corresponding linear parabolic problem { αvt a v xx =, v, x = u x + α u x as t +. Yang and Milani [] further extended the result of Nishihara [77] to any space dimension. In particular, for the linear damped wave equation.3., using Matsumura s estimates.3.4, they showed that ut vt L R n = Ot n/ t +, where u is the solution of.3. with u, u H [n/]+3 R n L R n H [n/]+ R n L R n and v is the solution of.3.3. In general, vt L does not decay faster than t n/. Therefore, the estimate above shows that the asymptotic profile of u is actually given by v. After that, Marcati and Nishihara [58] proved the following L p -L q decay estimates of the difference of u and v in the one-dimensional case n = :.3.9 ut vt e t/ u + t + u t/ L p Ct q p u, u L q for t and q p. The estimate above implies that u can be expressed asymptotically by ut, x vt, x + e t/ u x + t + u x t as t +. Here the term u x + t + u x t/ is the solution of the free wave equation w tt w xx = with the initial data u x,. Noting that in the estimate.3.4, we need some regularity on u, u, we can interpret that the term e t/ u x + t + u x t/

14 4. INTRODUCTION has the singularity of u. The estimate.3.9 was extended by Hosono and Ogawa [3] to n = and by Nishihara [78] to n = 3. More precisely, they obtained ut, vt, e t/ + t L W n tu + t W n tu + W n tu 8 p Ct n q p u, u L q for q p, where W n tg denotes the solution of the Cauchy problem of the free wave equation { wtt w =, t, x, R n, w, w t, x =, gx, x R n In particular, we obtain the following decomposition formula of S n t as t + : S n tg e t g + e t/ W n tg, where e t denotes the evolution operator of the heat equation. For higher dimensional cases, the corresponding L p -L q estimates were given by Narazaki [73]. He proved the following estimate for the lower frequency of the difference of solutions to the damped wave equation and the heat equation when n :.3. F {χξût ˆvt} L p C + t n q p +ε u, u L q, where q p, ε is an arbitrary small positive number and χξ is a compactly supported smooth radial function satisfying χξ = near ξ =. Moreover, in the case where < q < p <, p, q =, or p, q =,, we may take ε =. He also proved the following high frequency estimate F { χξût e t/ M t, û + M t, û } L p Ce δt u, u L q where δ >, < q p <, χξ is as above and M t, ξ = sint ξ k ξ /4 k! tk Θξ k k<n /4 k cost ξ k +! tk+ Θξ k+, M t, ξ = cost ξ + sint ξ k<n 3/4 k<n+/4 k<n /4 k k! tk Θξ k k k +! tk+ Θξ k+ + M t, ξ with Θξ = ξ ξ /4. We note that his proof is based on an argument using the Fourier transform and there is a loss ε of decay rate in.3.. Theorem.7 shows that the loss ε can be removable and in Chapter 8, we shall give a simpler proof based on the representation of the solution. Matsumura [6] also investigated when the energy of solutions to.3. decays to as t +. He proved that if at, x satisfies a + x + t at, x a

15 .3. A REVIEW OF SOME PREVIOUS RESULTS 5 with some a, a > and a t t, x, then the energy of the solution Et := u t t, x + ut, x dx R n tends to as t +. This result was extended by Mochizuki and Nakazawa [66]. They put the following condition on at, x:.3. a {e m + x + t loge m + x + t log [m] e m + x + t} at, x a with some a, a > and m Z, where e :=, e := e,..., e m := e em, log [] y := y, log [] y := log y,..., log [m] y := loglog [m ] y. They proved that if at, x satisfies.3. and a t t, x, then the energy of solution decays as u, u t t L C{log [m] e m + t} min{a/,}. On the other hand, for.3., Mochizuki [63] proved if n and at, x satisfies at, x C + x δ with some δ > and a t t, x is bounded and continuous, then the energy of solution does not decay in general. Moreover, he proved that the Møller wave operator exists and not identically zero. The scattering solution ut, x is asymptotically equivalent to a solution wt, x of the free wave equation in the following sense: lim u, u tt w, w t t L =. t For the case n =, we refer the reader to [7]. When n 3, this result was also extended by [66] to at, x satisfying at, x a {e m + x loge m + x log [m ] e m + x [ ] γ } log [m] e m + x with some a >, γ > and m Z. The energy decay problem in general exterior domain Ω R n has been investigated for a long time. It is well known that for the wave equation without damping, if Ω is nontrapping, then the local energy E R t := u t t, x + ut, x dx Ω B R decays exponentially fast if n is odd and polynomially fast if n is even, where B R := {x R n x < R}, R >. This is reasonable because the energy propagates along the wave front and the motion in the bounded region stops after time passes. Shibata [] considered the initial-boundary value problem of the damped wave equation u tt u + u t =, t, x, Ω,.3. u =, t, x, Ω, u, u t, x = u, u x, x Ω,

16 6. INTRODUCTION where n 3 and Ω is smooth and bounded. Let R > be a constant such that Ω c B R. He proved the following estimate of the local energy: E R t + ut L Ω B R C + t n u H 4 Ω + u H 3 Ω, provided that supp u, u Ω B R. We note that his result does not require any geometrical condition of Ω. Because the trapped energy also decreases by virtue of the dissipation term u t. After that, Dan and Shibata [9] extended the result above to n and improved the estimate as E R t + ut L Ω B R C + t n u H + u L, provided that supp u, u Ω B R. Nakao [68] considered.3. with the localized dissipation axu t instead of u t, where ax for x > L with sufficiently large L > and ax c > with some constant c > on a neighborhood of the closure of.3.3 Γx = {x Ω νx x x > } with some x R n. Here νx denotes the unit outward normal vector of Ω at the point x Ω. We note that if Ω c is star-shaped with respect to x, then Γx is empty. He proved that if supp u, u B L, then the local energy decays as E L+εt t C ε,δ + t +δ with arbitrary < ε, δ <. Moreover, for the case of odd dimensions, the local energy decays exponentially. This result says that we need only the dissipation on a part of Ω for the local energy decay. Matsuyama [6] considered a dissipation depending on time and space variables at, xu t and removed δ in the above rate and relaxed the assumption on at, x as at, x a {e m + x loge m + x [log [m] e m + x ] γ } with some a >, γ > satisfying a < γ γ, instead of the condition a for large x. Moreover, under some suitable additional assumptions on at, x and the initial data, he proved that the total energy of solutions does not decay in general and the solution is asymptotically free as t +. For the total energy decay of solutions to u tt u + axu t =, t, x, Ω, u =, t, x, Ω, u, u t, x = u, u x, x Ω with an exterior domain Ω R n has been also considered by Nakao [69] see also [7]. He put the following assumptions on ax: i ax c > holds on some neighborhood of the closure of Γx with some c > and x R n. Here Γx is defined by.3.3. ii ax c > for all x L with some c, L >. Under these assumptions, he proved ut L C u H + u L, Et CE + t, where Et is the total energy of u, that is, Et := u t t, x + ut, x dx. Ω

17 .3. A REVIEW OF SOME PREVIOUS RESULTS 7 When Ω c includes the origin and is star-shaped with respect to the origin and ax c > holds for all x L with some c, L >, Ikehata [8] improved the decay rates in the above estimates as where ut L C + t u, u H L + d u + a u L, Et C + t u, u H L + d u + a u L, dx = { x n 3, x logb x n = with some constant B > satisfying B inf x Ω x. Ikehata [3] considered the special dissipative term a x α u t with α <. He obtained the boundedness of the weighted energy e a x α α t u t t, x + ut, x dx C. Ω Todorova and Yordanov [7] improved the above estimate by introducing a new weighted energy method. They considered the Cauchy problem for the wave equation with space-dependent damping { utt u + axu.3.4 t =, t, x, R n, u, u t, x = u, u x, x R n. They assumed that ax a + x α with some α [, and there exists a solution of the Poisson equation.3.5 Ax = ax having the following properties: a a a3 Ax, Ax = O x α as x +, axax ma := lim inf x Ax >. It is known that such solutions Ax exist if ax is radially symmetric and satisfies a + x α ax a + x α with some a, a > and α [,. Using the function Ax, they constructed a weight function of the form e ma εax/t and obtained the following weighted energy estimates:.3.6 e ma εax/t axut, x dx C ε u L + u L t ma+ε, R n.3.7 e ma εax/t u t t, x + ut, x dx R n C ε u L + u L t ma +ε

18 8. INTRODUCTION for large t > and any ε >, provided that the data u, u have compact support. In particular, if ax is radially symmetric and behaves like ax a x α as x +, then it follows that a Ax αn α x α as x +, and hence, ma = n α α e ψt,x axut, x dx C ε u L + u L t R n n α α +ε, e ψt,x u t t, x + ut, x dx C ε u L + u L t R n for large t > and any ε >, where ψt, x = a x α α + ε t. Their method is also applicable to the corresponding heat equation axv t v = n α α +ε and we can obtain the same decay rate. This indicates that in this case the equation.3.4 still has the diffusive structure. However, the precise asymptotic profile was remained as an open problem. Recently, Nishiyama [88] proved the diffusion phenomenon for the abstract damped wave equation u + Au + Bu =. His result includes space-dependent damping which does not decay near infinity. Due to the authors knowledge, Theorem. stated in the previous section is the first result for the precise asymptotic profile of solutions to the damped wave equation with space-dependent decaying potential. We also mention that the above result by Todorova and Yordanov was extended to damping depending on time and space variables at, x = axbt satisfying a + x α ax a + x α, b + t β bt b + t β with α [,, β [,, α + β [, by J. S. Kenigson and J. J. Kenigson [45]. Recently, Ikehata, Todorova and Yordanov [37] considered the critical case a x ax a x. They obtained several optimal energy estimates of solutions with compactly supported data. More precisely, they proved u t, ut L = Ot mina,n+ε as t +, where ε is arbitrary small positive number. When n 3, < a < n or n =,, n < a, we can remove ε in the above estimate. Moreover, if ax is radially symmetric, ax a x as x and < a < n, then the decay rate a / in the above inequality is optimal. We note that when a n, the decay rate n/ agrees with that of the corresponding heat equation. Therefore, we expect that this decay rate is also optimal. However, this optimality is still open and the asymptotic profile of solutions is completely open as far as the author s knowledge.

19 .3. A REVIEW OF SOME PREVIOUS RESULTS 9 For the time dependent damping cases, more specific asymptotic behavior of solutions was investigated by Wirth [4, 5, 6, 7, 9]. He considered the linear damped wave equation.3.8 u tt u + btu t =. For simplicity, we assume that bt is positive, smooth, monotone and satisfies d k dt k bt C k + t k bt for k Z. A typical example of bt is + t β with β R. We also consider the free wave equation.3.9 w tt w = and the corresponding heat equation.3. btv t v =. We denote by λt = exp t bsds an auxiliary function. He determine the behavior of solutions to.3.8 as time tends to infinity in the following five cases: i scattering If bt L,, then the solution to.3.8 is asymptotically free. More precisely, there exists an isomorphism W + on Ḣ R n L R n such that for the solution ut, x to.3.8 with initial data u, u and the solution wt, x to.3.9 with the initial data W + u, u, the asymptotic equivalence lim u, u tt w, w t t L = t holds. ii non-effective dissipation If lim sup t tbt <, then the solution u to.3.8 satisfies the L p -L q estimate u, u t t L p C n + t q p u λt W s+,q + u W s,q for p [,, q is the dual of p and s > n/q /p. Moreover, λtu is asymptotically free in the sense that there exists a solution w of.3.9 satisfying lim λt u, u tt w, w t t L =. t iii scale-invariant weak dissipation If bt = µ/ + t with µ >, then the solution u of.3.8 satisfies the L p -L q estimate u, u t t L p n max{ C + t q p µ, n q p } u W s+,q + u W s,q for p [,, q is the dual of p and s > n/q /p. iv effective dissipation If tbt + as t +, then the solution u of.3.8 satisfies the L p -L q estimate u, u t t L p C + t n bs q p ds u W s+,q + u W s,q

20 . INTRODUCTION for p [,, q is the dual of p and s > n/q /p. Moreover, if bt 3 / L,, then the lower frequency part of the solution u of.3.8 is asymptotically equivalent to that of a solution v of.3. in the L -sense. This is called the local diffusion phenomenon. It is also follows that when bt 3 L,, u is for each frequency asymptotically equivalent to that of v. This is called the global diffusion phenomenon. v overdamping If bt L,, then the solution u of.3.8 with data from L R n H R n converges as t to the asymptotic state u, x = lim t ut, x in L R n. Furthermore, this limit is non-zero for non-zero initial data. He also treated in [9] the time periodic dissipation, that is, bt+t = bt > for t with some T >, and proved that Matsumura s estimates are still true in this case. We remark that for the time-dependent speed and damping case u tt at u + btu t =, recently, D Abbicco and Ebert [6] gave an extension of the results i and ii above. We also mention the abstract damped equation.3. { u + u + Au =, u = u, u = u and the corresponding the heat equation.3. { v + Av =, v = u + u in a separable Hilbert space H. Here A is a closed, self-adjoint and nonnegative operator on H with a dense domain DA. The diffusion phenomenon for the abstract equation.3. is closely related to the problem on exterior domains. Ikehata [7] considered the concrete case H = L Ω, A =, DA = H Ω H Ω with an exterior domain Ω R n having compact smooth boundary, and proved for solutions u to.3. and v to.3. that ut vt L Ω = O t log t as t +. After that, Ikehata and Nishihara [3] considered abstract case and improved the above estimate as ut vt H = Ot log t /+ε with arbitrary small ε >. They conjectured the optimal rate is given by Ot. Chill and Haraux [3] solved this conjecture. They proved that.3.3 ut vt DA / Ct u DA / + u H for t, where DA / denotes the graph norm of A / note that this operator is well-defined: u DA / := A/ u H + u H.

21 .3. A REVIEW OF SOME PREVIOUS RESULTS Moreover, they proved the decay rate above is optimal. Radu, Todorova and Yordanov [97] obtained the following estimates, which are stronger than.3.3: ut vt H Ct e ta/ u H + e ta/ u H + Ce t/6 u H + A / + u H, A k ut vt H Ct k e ta u H + e ta u H + Ce t/6 A k u H + A / + A k u H for any t and k. These estimates allow us to transfer the decay from the heat equation to the hyperbolic equation. They applied these estimates to operators A generating a Markov semigroup on L Ω, µ with some σ-finite measure space Ω, µ. They obtained faster decay of the difference ut vt with L data and they also proved an abstract version of Matsumura s estimates.3.4. Recently, Ikehata, Todorova and Yordanov [38] proved similar estimates for the strongly damped wave equation u + Au + Au =. Yamazaki [] extended the results of Chill and Haraux [3] to time-dependent damping cases: { u btu + Au =, u = u, u = u and.3.5 where bt is a C function satisfying and { btv + Av =, v = v, b + t β bt b + t β, < β <, b t b + t β v = u + u b u b s bs exp s bσdσ ds. Let k, l and u, u DA k+/ RA l DA k RA l, where RA l denotes the range of A l. Then she proved that A k ut vt DA / Ct β +βk+l u DA k+/ + ũ H + u DA k + ũ H, A k ut vt H Ct β +βk+l u DA k+/ + ũ H + u DA k + ũ H for t, where DA k+/, DAk are the graph norm of A k+/, A k, respectively, and ũ, ũ are elements of H such that A l ũ = u, A l ũ = u, respectively. Wirth [8] treated the critical case bt b + t. He proved that if bt satisfies b t b + t, b / L,, lim sup t tbt < and ker A = {}, then for the solution ut of.3.4, it holds that lim λtut, t u t wt, w t E =, where wt is the a solution of the free wave equation w + Aw =,

22 . INTRODUCTION λt = exp t bsds and ϕ, ψ E is the energy norm ϕ, ψ E = A/ ϕ H + ψ H. Moreover, the operator mapping u, u to w, w is injective..3.. Damped wave equations with nonlinear source terms. We consider the Cauchy problem for the semilinear damped wave equation { utt u + at, xu.3.6 t = fu, t, x, R n, u, u t, x = u, u x, x R n, where fu denotes the nonlinear term. Here we treat source semilinear terms, namely fu = f u = u p u or f u = u p with p >. First we mention the constant coefficient case at, x. In this case, for the corresponding semilinear heat equation { vt v = fu, t, x, R.3.7 n, v, x = v x, x R n, there are many literature on the structure of solutions. known that there is the critical exponent In particular, it is well p F = + n dividing the behavior of solutions into the following way see [3, 5, 5, ]: iif p > p F, then for any small data v L R n L R n there is a unique global solution v of.3.7 satisfying vt, x θ Gt, x as t +, where θ = v xdx + fvt, xdxdt. R n R n iiif p p F, then for the data v satisfying v and v, the locally-intime solution vt, x blows up in finite time, that is, lim vt, x = + t T for some x R n and T >. Moreover, for the data εv with small parameter ε > and fixed v, the lifespan of the solution is estimated as.3.8 T ε with T ε = sup{t, ] vt, x < + for t [, T ]} { e Cε p p = p F, Cε /κ < p < p F κ = p n and some C >. From the viewpoint of the diffusion phenomenon, it is expected that for the damped wave equation.3.9 { utt u + u t = fu, t, x, R n, u, u t, x = u, u x, x R n,

23 .3. A REVIEW OF SOME PREVIOUS RESULTS 3 the same results hold. For f u = u p u, existence of classical solutions has been investigated for a long time see [99, 59, 6]. For weak solutions, it is well known that if u, u H R n L R n and < p n/n < p < if n =,, then there are some T > and a unique solution u C[, T ; H R n C [, T ; L R n. Moreover, the finite propagation speed property holds, that is, if with some L >, then it is true that supp u supp u {x R n x < L} supp ut {x R n x < t + L} see [, 39]. Levine [5] proved that if the initial data satisfy u L + u L p + u p+ L <, p+ then the local solution blows up in finite time. This result shows that the existence of global solution requires some smallness condition on the data. Nakao and Ono [67] proved the existence of global solutions with suitably small and compactly supported data in H R n L R n when + 4 n p < n + n. Their proof was based on the so-called modified potential well method which was originally introduced by Payne and Sattinger [94]. When n 3, Ikehata, Miyaoka and Nakatake [3] proved that if the initial data u, u H L L L are sufficiently small and + n < p < n =,, < p n = 3, n n then there exists a unique global solution satisfying the decay estimates ut L C + t n/4, u t, ut L C + t n/4 /. Their method is also applicable for f u = u p. We note that there is a gap between the exponent p = and the critical exponent p = p F = + /3 when n = 3. Moreover, in view of Matsumura s estimate.3.4, we can expect that the decay rate of u t t L is faster than that of ut L. On the other hand, Li and Zhou [53] obtained small data blow-up results for f and f when n =, and < p + /n. These results show that when n =,, the critical exponent for.3.9 with the nonlinearity f and f is actually given by p F = + /n. After that, Nishihara [78, 79] determined the critical exponent for.3.9 with f, f when n = 3. He proved that if the initial data u, u W, W, L L are sufficiently small and p > + /3, then the Cauchy problem.3.6 admits a unique global solution u C[, ; L L C[, ; H C [, ; L satisfying the decay estimates ut L q C + t 3 q

24 4. INTRODUCTION for q and ut L C + t 3/4 /, u t t L C + t 3/4. Taking into account the linear estimates described in the previous subsection, we can expect the above decay estimates are optimal. When n =, Hosono and Ogawa [3] obtained the corresponding results. They proved that if the initial data u, u B, W, B, L are sufficiently small and p >, then there exists a unique global solution of.3.6 satisfying for q and u C[, ; H L C [, ; L ut L q C + t q ut L C + t. Here B, s denotes the Besov space defined by B,R s n := f : Rn R; f B s, := js ϕ j f L < +, j where {ϕ j } is the Littlewood-Paley dyadic decomposition see [] for detail. Ono [9] relaxed the assumption on the data of [78] to u, u H L L L and proved that the problem.3.6 admits a unique global solution u C[, ; H L C [, ; L satisfying.3.3 ut L q C + t n q,.3.3 ut L C + t n 4 /,.3.3 u t t L C + t n 4 for q when n 3 and p > + /n. For n = 4, 5, Narazaki [73] obtained the global existence results for f, f. He proved that if p F < p n/n, p < and the initial data satisfy u, u H W,p/p W,p L H L p/p L p L and sufficiently small, then there exists a unique global solution u C[, ; H L p/p L p C [, ; H C [, ; L satisfying the estimates for p q p/p. Ono [93] obtained the estimate.3.3 for q n/n under the assumption u, u H W, L W,. For higher dimensional cases with f u = u p, Todorova and Yordanov [5, 6] developed a new weighted energy method and proved the existence of global solutions with small and compactly supported data when p F < p n/n p F < p < when n =, and local solution blows up in finite time if the initial data satisfy R n u i xdx >

25 .3. A REVIEW OF SOME PREVIOUS RESULTS 5 for i =, and < p < p F. After that, Qi S. Zhang [] and Kirane and Qafsaoui [47] proved that the critical exponent p = p F belongs to the blow-up case. In particular, Kirane and Qafsaoui [47] obtained blow-up results for more general semilinear dissipative wave equation of the form u tt + m x α m u + u t = ft, x u p + wt, x, where m, p >, ft, x satisfies fr t, R /m x CR λ for large R > with some λ, α < mλ +, and wt, x satisfies x α wt, x L [, R n and x α wt, xdxdt. In this case, they proved that if < p + mλ + α/n and x α u R n + u dx >, then there is no global-in-time solution. The method by Todorova and Yordanov [6] for proving the global existence result is a weighted energy estimate by using a weight function of the form e ψt,x. They pointed out the following identity: [ ] e e ψ ψ u t u tt u + u t = t u t + u e ψ u t u + eψ ψ t u u t ψ + e ψ + ψ + ψ t u t. ψ t ψ t They chose the weight function ψ so that ψ t <, ψ t + ψ ψ t =. More precisely, they put ψt, x = t + ρ t + ρ x with ρ >. To make sense the definition of ψ, they need the compactness of the support of the data, that is, supp u, u {x R n x ρ}. Ikehata and Tanizawa [35] chose the function ψ as ψt, x = x 4 + t and succeeded to remove the assumption on the compactness of the data. For the estimate of the lifespan in the critical and subcritical cases < p p F, Li and Zhou [53] and Nishihara[79] obtained the same estimate as.3.8 when n =, and n = 3, respectively. Theorem.6 described in the previous section shows when the subcritical case < p < p F, the same estimate as.3.8 holds for solutions to.3.9 with f = f. However, to estimate the lifespan in the critical case p = p F with n 4 remains open. The asymptotic profile of the global solution in supercritical cases p > p F was investigated by Nishihara [78], Hosono and Ogawa [3] and Hayashi, Kaikina and Naumkin [6]. Nishihara [78] and Hosono and Ogawa [3] proved that for suitably small data, the corresponding solutions behave as ut θgt L q = ot n q for q when n = 3 and n =, respectively. Here and Gt, x = 4πt n/ e x /4t θ = u + u xdx + R n R n fudxdt.

26 6. INTRODUCTION Hayashi, Kaikina and Naumkin [6] extended these results to any space dimensions. Let H l,m be a weighted Sobolev space defined by H l,m = {ϕ L x m i x l ϕ L < }. They proved that for f and f, if p > p F and the initial data belong to.3.33 u H α, H,δ, u H α, H,δ and sufficiently small, where δ > n, [α] < p; α n p for n and α [ p, for n =, then there exists a unique solution satisfying u C[, ; H α, H,δ.3.34 ut θgt, x L q Ct n q min, δ n 4, n p for t > and q [, n/n α] α < n/, q [, α = n/, q [, ] α > n/. We note that their result also does not require the compactness of the support of the data. Kawakami and Ueda [43] obtained more precise asymptotic expansion of solutions to.3.6 when n 3. For k and l Z, we define ϕ L k = + x k ϕx dx, R n ϕ W l, k = α l α x ϕ L k and L k = {ϕ L ϕ L k < + }, W l, k = {ϕ W l, ϕ W l,. They proved that if u, u W, W, k L L k and k u C[, ; L L, ; L is a solution of.3.6, then it follows that t n q ut V t L q { Ot k/ + Ot n p + Ot n = p k, Ot k/ log t + Ot n p = k as t + for q [, ]. Here V t := α [k] M α vt, tg α t, x, where g α t, x := α α! x α G + t, x, M vt, t := vt, xdx, R n M α vt, t := x α vt, xdx α =, R n M α vt, t := x α vt, xdx R n M β vt, t x α g β t, xdx α, β<α R n and vt, x is the solution of the inhomogeneous linear heat equation { vt v = fu, t, x, R n, v, x = u x + u x, x R n.

27 .3. A REVIEW OF SOME PREVIOUS RESULTS 7 We also mention the critical exponent for.3.6 with initial data not belonging to L R n. Ikehata and Ohta [34] proved that if the initial data belong to.3.35 u, u H L m L L m and + m n < p < n =,, + m n < p n 3 n 6, n where m lies in n + 6n n n m n =,, m < min, 3 n 6, 4 n then there exists a unique global solution u C[, ; H C [, ; L of.3.6 satisfying ut L C + t n m, u t, ut L C + t n m /. Moreover, they also showed the nonexistence of global solution even for small data in the subcritical case < p < +m/n for all n. This indicates that the critical exponent of.3.6 for the data belonging to.3.35 is given by p c = + m/n at least when n 6. We expect that this is also true for higher dimensional cases. After that, Narazaki and Nishihara [75] considered the initial data satisfying u, u = O x kn as x + with some k, ]. For the corresponding heat equation.3.7, Lee and Ni [5] determined the critical exponent as p c = + kn for the data satisfying v = O x kn as x +. Narazaki and Nishihara [75] obtained the asymptotic profile of the solution v of.3.7 with the initial data satisfying v CR n, x kn v x L and lim x x kn v x = c. They proved that the profile of v is given by V t, x = c Gt, x y y R kn dy, where n G denotes the heat kernel G = 4πt n/ e x /4t. They also proved that if the initial data u, u C [n/] R n CR n satisfies x kn x α u L α [n/], x kn u L and p > + /kn, then the problem.3.6 admits a unique global solution u C[, R n satisfying x kn ut, x L R n for all t. Moreover, if the initial data satisfy lim x x kn u, u x = c, then the asymptotic profile of u is also given by V t, x defined above. Next, we consider the variable coefficient cases. Ikehata, Todorova and Yordanov [36] considered the semilinear wave equation with space-dependent damping { utt u + axu.3.36 t = u p, t, x, R n, u, u t, x = u, u x, x R n. Under the assumption that ax is radially symmetric and ax a x α as x +, they proved that the critical exponent is given by p c = + n α

28 8. INTRODUCTION for compactly supported initial data the global existence part is also true for f u = u p u. This exponent agrees with that of the corresponding heat equation. The proof of global existence part is based on a weighted energy method by using the weight function wt, x = t ma/ ε e ma/ ε/ax/t, where ε is arbitrary positive number, Ax is the positive solution of the Poisson equation.3.5 satisfying a-a3 and ma is defined by a3. The proof of the blow-up part is done by the test function method developed by Qi S. Zhang []. This method requires the positivity of the nonlinearity and is not applicable to f u = u p u. We mention that Theorem.6 gives an estimate of the lifespan from above. For the time-dependent damping case { utt u + btu.3.37 t = u p, t, x, R n, u, u t, x = u, u x, x R n with bt = b + t β, < β <, Nishihara [83] and Lin, Nishihara and Zhai [56] determined the critical exponent as p c = + n, which also coincides with that of the corresponding heat equation.3.38 btv t v = v p. Note that for.3.38, by changing the variable as s = Bt := t bs ds, the equation.3.38 is reduced to ut, x = ϕs, x = ϕbt, x, ϕ s ϕ = ϕ p. The global existence part was proved by a weighted energy method by using a weight function wt, x = Bt n/4+ε/ e x /4+δBt, where ε is arbitrary small positive number and δ is a suitably chosen small positive parameter. To obtain the blow-up result, they introduced a modified test function method. By multiplying the equation.3.37 by a nonnegative function gt C [,, it follows that gtu tt gu g tu t + g t + btgtu = gt u p. They chose gt by the solution of the Cauchy problem of the ordinary differential equation { g t + btgt =, g = exp t bsds dt. Then we can obtain the equation of the divergence form and so we can apply the test function method by Zhang []. D Abbicco and Lucente [7] applied this method to more general dissipative wave equations with time-dependent coefficients. This idea is also based on Theorem.5 in which we consider the one-dimensional semilinear wave equation with damping depending on time and space variables. D Abbicco,

29 .3. A REVIEW OF SOME PREVIOUS RESULTS 9 Lucente and Reissig extended the global existence result in the above one [56] to more general effective damping. The author [] considered the critical exponent problem for the critical case bt = µ +t with µ > and proved that if p > p F and µ is sufficiently large depending on p, then there exists a unique global solution for the small data decaying very fast near the infinity see Theorem.3. After that, D Abbicco [5] improved the global existence result for µ n + and p > p F. For the semilinear damped wave equation with time and space dependent coefficients.3.39 { utt u + a x α + t β u t = u p, t, x, R n, u, u t, x = u, u x, x R n, the author [9] proved the small data global existence of solutions under the assumption α, β, α + β <, p > + n α, provided that the initial data decay sufficiently fast near the infinity see Theorem.. Recently, a similar result is obtained by Khader [46]. The exponent above is expected to be critical. However, this problem is still open. The difficulty is that when the damping depends on time and space variables, we cannot apply the test function method directly. The author [] proved that in the one dimensional case, if the damping term decays sufficiently fast as t +, then we can transform the equation into divergence form and apply the test function method see Theorem Damped wave equations with nonlinear absorbing terms. Here we consider the semilinear damped wave equation with absorbing nonlinearity { utt u + u.3.4 t = u p u, t, x, R n, u, u t, x = u, u x, x R n. In this case, there is a unique global solution any large data in L R n. For the corresponding heat equation { vt v = v.3.4 p v, t, x, R n, v, x = v x, x R n, it is known that the asymptotic behavior of solutions divide in the following way: p > p F vt, x θ Gt, x, p = p F vt, x θ log t n/ Gt, x, p < p F vt, x w b t, x = t /p f b x/ t, where p F = + /n, Gt, x = 4πt n/ e x /4t, θ = v xdx v p vdxdt R n R n and w b t, x b is the self-similar solution with the profile f b t, x satisfying r f + n f + f p f = r p f, lim r r/p fr = b.

30 3. INTRODUCTION For.3.4, it is expected that the same results hold. Kawashima, Nakao and Ono [44] proved that if u, u H L, then there exists a unique global solution u C[, ; H C [, ; L for.3.4. Moreover, if u, u L and p > + 4/n, then the solution decays as ut L C + t n/4 for n 4. Using this result, Karch [4] proved ut θ Gt L = ot n/4 as t + when p > + 4/n. After that, Nishihara and Zhao [87], Nishihara [8] and Ikehata, Nishihara and Zhao [33] improved the results above as follows: ilet u, u H L satisfy where u, u, u, u p+/ H,m, m = p n δ with some δ >. We assume that m > n if p > + /n. If < p < n/[n ] + and p + 4/n, then the solution decays as.3.4 ut L q C + t /p +n/q for q, q <, q n/n when n =, n =, n 3, respectively. iiif n 3 resp. n = 4, u, u H H u, u H,m and p F < p + 4/n resp. p F < p < + 4/n, then it follows that ut θ Gt, x L q = ot n /q for q, where θ = u + u xdx u p udxdt. R n R n The decay rate in.3.4 agrees with that of the self-similar solution w b t, x. Thus, we expect this rate is optimal in the subcritical case < p < p F. Hayashi, Kaikina and Naumkin [6, 7, 8, 9,, ] and Hayashi, Naumkin and Rodriguez-Ceballos [] proved several results on the asymptotic profile of solutions. In [6], it is proved that for the initial data satisfying.3.33, the solution u satisfies.3.34 the condition on α, δ, p are the same as before. This result shows that our expectation in the supercritical case is true. In the subcritical case with n = and p 3 ε, 3 for some small ε >, in [8] they obtained the following asymptotic behavior: for the small data satisfying with a,, where γ = integral equation V ξ = e ξ /4 4π / ut, x tη /p V x/ t + Ot /p γ u, + x u L L,a L L,a mina, p, V L L,a is the solution of the η4π / e ξ y z /4 z F ydydz, z z / R