u i t=0 = u i0 (x) 0, (1.2)

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1 Electronc Journal of Dfferental Euatons, Vol. 8 (8), No. 9, pp. 3. ISSN: URL: or NONEXISTENCE OF GLOBAL SOLUTIONS TO THE SYSTEM OF SEMILINEAR PARABOLIC EQUATIONS WITH BIHARMONIC OPERATOR AND SINGULAR POTENTIAL SHIRMAYIL BAGIROV Communcated by Ludmla S. Pulkna Abstract. In the doman Q R = {x : x > R} (, ) we consder the problem u t u C x u = x σ u, u t= = u (x), u t u C x u = x σ u, u t= = u (x), Z Z Z Z u ds dt, u ds dt, B R B R where σ R, >, C < ( n(n ) ), =,. Suffcent condton for the nonexstence of global solutons s obtaned.the proof s based on the method of test functons.. Introducton Let us ntroduce the followng notaton: x = (x,..., x n ) R n, n >, r = x = x x n, B R = {x; x < R}, B R = {x; x > R}, B R,R = {x; R < x < R }, Q R = B R (; ), Q R = B R (; ), B R = {x; x = R}, u = ( u x,..., u x n ), C, x,t (Q R ) s the set of functons that are four tmes contnuously dfferentable wth respect to x and contnuously dfferentable wth respect to t n Q R. In the doman Q R we consder the system of euatons wth the ntal condton u t u C x u = x σ u u t u C x u = x σ u, (.) u t= = u (x), (.) Mathematcs Subect Classfcaton. 35A, 35B33, 35K5, 35K9. Key words and phrases. System of semlnear parabolc euaton; bharmonc operator; global soluton; crtcal exponent; method of test functons. c 8 Texas State Unversty. Submtted November, 7. Publshed January 6, 8.

2 S. BAGIROV EJDE-8/9 and the condtons B R u dx dt, B R u dx dt, (.3) where n >, >, σ R, C < ( n(n ) ), u (x) C(B R ), u = ( u), u = n u x k k=, =,. We wll study the nonexstence of a global soluton of problem (.)-(.3). By a global soluton of problem (.)-(.3) we understand a par of functons (u, u ) such that u (x, t), u (x, t) C, x,t (Q R ) C3, x,t (B R (; )) C(B R [; )) and satsfy the system (.) at every pont of Q R, the ntal condton (.) and condtons (.3). The problems of nonexstence of global solutons for dfferental euatons and neualtes play a key role n theory and applcatons. Therefore, they have a constant attenton of mathematcans, and a great number of works were devoted to them [,, 3,, 9,, 3, 6,, ]. A survey of such results can be found n the monograph [7]. In the classcal paper [7], Futa consdered the followng ntal value problem u t = u u, (x, t) R n (, ), u t= = u (x), x R n, (.) and proved that postve global solutons of problem (.) do not exst for < < = n. If >, then there are postve global solutons for small u (x). The case = was nvestgated n [, ] and t was proved that n ths case there are no postve global solutons. Pnsky [9] showed the exstence and nonexstence of global solutons n R n (, ) to the euaton u t u = a(x)u, where > and a(x) behaves lke x σ wth σ > for large x. The results of Futa s work [7] aroused great nterest n the problem of the nonexstence of global solutons, and they were expanded n several drectons. For example, varous bounded and unbounded domans were consdered nstead of R n, as well as more general operators than the Laplace operator ncludng dfferent type nonlnear operators were consdered (for more comprehensve treatment of such problems, see [, 7, ] and references there n). Another may of extendng of Futa s result s to nvestgate a system of Futatype reacton-dffuson euatons, and ths s what we do here. For example, many authors have nvestgated the exstence and nonexstence of global and local solutons to the ntal value problem u t = α u t k x σ v, u t= = u (x) (.5) v t = α v t k x σ u, v t= = v (x). Escobedo and Herrero [5] consdered problem (.5) on R n (, ) wth α =,k =, σ =, >, >, =, and proved that f max(, ) n, then for any nontrval ntal functons there are no nonnegatve global solutons. Fla, Levne and Uda [6] consdered problem(.5) on R n (, ) wth α, α =, k =, σ =,, >, =, and studed the exstence of nonnegatve global and non-global solutons. In the case α =, k =, =,,

3 EJDE-8/9 NONEXISTENCE OF GLOBAL SOLUTIONS 3 Mochzuk and Huang [8] proved the exstence and nonexstence theorems for global solutons and studed asymptotc behavor of the global soluton of problem (.5) on R n (, ). Carst [8] consdered problem (.5) for k, σ R,, > on R n (, ), and nonexstence of global soluton s studed. Levne [5] studed nonnegatve solutons of the ntal boundary value problem for the system of euatons n (.5) for α =, k =, σ =, =, n doman D (, ), where D s a cone or the exteror of a bounded doman. In the present paper we consder a system of semlnear parabolc euatons wth bharmonc operator and sngular potental n the exteror doman Q R. Usng the technue of test functons worked out by Mtder and Pohozaev n [6],[7], we fnd a suffcent condton for nonexstence of global nontrval soluton.. Man result and ts proof The avod complcatons, we ntroduce the followng denotaton: D = ( (n ) C, λ ± n ) = ± D, µ = ( D λ ) λ, µ = ( D λ ) λ, α = λ σ n λ n β = λ σ n λ n θ = σ (σ ) θ = σ (σ ) Let us consder the functons, α = λ σ n λ n,, β = λ σ n λ n, λ n, λ n, =,. ξ (x) = µ x n λ µ x n λ x n λ, =,. It s easy to verfy that ξ (x) are the soluton of the euaton n R n \{} and for x =, u C x u = (.) ξ =, ξ r = D, ξ =, The man result of ths paper reads as follows. ( ξ ) r. (.) Theorem.. Assume that n >, β >, C < ( n(n ) ) and < β, max(θ, θ ), (, ) (α, β ) n case α >, (, ) (β, α ) n case α >, =,. Then there s no nontrval global soluton of (.)-(.3).

4 S. BAGIROV EJDE-8/9 Proof. For smplcty we take R =. Assume that (u (x, t), u (x, t)) s a nontrval soluton of (.)-(.3). Let us consder the followng two functons:, for x ρ, ϕ(x) = ( x ρ )κ, for ρ x ρ, for x ρ,, for t, T ρ (t) = ( ρ t) γ, for t, for t, where κ, γ are large postve, and κ s such number that for x = ρ, ϕ = ϕ r = ϕ r = 3 ϕ =. (.3) r3 We multply the frst euaton by ψ (x, t) = T ρ (t)ξ (x)ϕ(x), the second by ψ (x, t) = T ρ (t)ξ (x)ϕ(x) and ntegrate over. After ntegraton by parts, we obtan the followng relatons = x σ u T ρ (t)ξ (x)ϕ(x) dx dt B,ρ u ξ ϕ dt ρ dt dx dt u T ρ (ξ ϕ) dx dt C x u T ρ ξ ϕ dx dt u (x)ξ (x)ϕ(x)dx B [ ( u ) (ξ ϕ) T ρ (t)dt ξ ϕds u ds B,ρ ν B,ρ ν u ν (ξ ] ϕ)ds u ν (ξ ϕ)ds, B,ρ (.) where ν s a unt vector of external normal to B, ρ,, =,,. In order not to be repeated, n what follows, we wll take nto account that, =,, and n all expressons wll wrte the same constant C, but n fact, n each expresson C ndcates dfferent constants. Usng (.), (.3), we estmate the ntegrals n suare brackets n (.). B,ρ B,ρ (ξ ϕ) u ds = ν ( u ) ξ ϕds =, ν = = x = x = x = (ξ ϕ) u ds ν u ( ξ r ϕ ξ ϕ r )ds ξ u ds, r

5 EJDE-8/9 NONEXISTENCE OF GLOBAL SOLUTIONS 5 Snce B,ρ u ν (ξ ϕ)ds = u B,ρ B B,ρ u ν ( ξ ϕ ( ξ, ϕ) ξ ϕ)ds u = x = r ξ ds =, ν ( (ξ ϕ))ds = x = = u (x)ξ (x)ϕ(x)dx, x = and u ν ( ξ ϕ)ds ( ξ ) u ds. r T ρ (t)dt, takng nto account that ξ s the soluton of n (.) and usng the above estmates, from (.) we obtan x σ u T ρ (t)ξ (x)ϕ(x) dx dt = n k,m= ρ u ξ ϕ dt ρ dt u ξ ϕ dt ρ dt dx dt dx dt u T ρ C (ξ ϕ) dx dt Q x u T ρ ξ ϕ dx dt u T ρ ϕ( ξ C x ξ ) dx dt u T ρ [( ( ξ ), ϕ) ( ξ, ( ϕ)) ξ ϕ ξ x k x m B u ξ ϕ dt ρ dt ϕ ] dx dt x k x m ρ dx dt u T ρ H (ξ, ϕ) dx dt, B ρ,ρ where H (ξ, ϕ) denotes the expresson n the suare brackets,.e. H (ξ, ϕ) = ( ( ξ ), ϕ) ( ξ, ( ϕ)) ξ ϕ n ξ ϕ. x k x m x k x m k,m= (.5) (.6) Usng the Holder s neualty, we estmate the rght-hand sde of (.5). We can wrte: x σ u T ρ ξ ϕ dx dt ( ( B ) / x σ u T ρ ξ ϕ dx dt B dtρ dt ξ ϕ T ρ x σ( ) ξ ) / dx dt

6 6 S. BAGIROV EJDE-8/9 ( x σ u T ρ ξ ϕ dx dt B ρ,ρ ( H (ξ, ϕ) Bρ,ρ Tρ ) / dx dt x σ( ) ξ ϕ ) / where =. Let us denote the second ntegral n the frst addend above by I, and the second ntegral n the second addend by J. If we wrte separately, then from (.6) we obtan the followng: x σ u T ρ ξ ϕ dx dt ( ( (.7) /[ x σ u / T ρ ξ ϕ dx dt) I ] J /, x σ u T ρ ξ ϕ dx dt /[ x σ u / ρξ ϕ dx dt) I J Usng (.6), from these neualtes we obtan x σ u T ρ ξ ϕ dx dt [( B x σ u T ρ ξ ϕ dx dt) /I / ( x σ u T ρ ξ ϕ dx dt B ρ,ρ x σ u T ρ ξ ϕ dx dt [( B ) /J / x σ u T ρ ξ ϕ dx dt) /I /, ]. (.8) ] /[ / I ] J /, ( ] x σ u T ρ ξ ϕ dx dt) / J /[ / I ] J /. B ρ,ρ Substtutng (.8) n (.7) and (.7) n (.8), we obtan x σ u T ρ ξ ϕ dx dt ( ( ) x σ u [ / T ρ ξ ϕ dx dt I ][ J / / I ] J / /, x σ u T ρ ξ ϕ dx dt ) x σ u [ / T ρ ξ ϕ dx dt I ][ J / / I ] J / /. (.9) (.)

7 EJDE-8/9 NONEXISTENCE OF GLOBAL SOLUTIONS 7 Hence x σ u T ρ ξ ϕ dx dt ] [ I / J [ / ] I J /, x σ u T ρ ξ ϕ dx dt Makng the substtutons [ ] I / J / [ ] I / J /. t = τ, r = ρs, x = ρy, T (τ) = Tρ ( τ), ξ (y) = ξ (ρy), ϕ(y) = ϕ(ρy), (.) (.) (.3) we estmate the rght-hand sdes of (.) and (.). Frst, we estmate the ntegrals I, =,. where I = ρ ρ dt ρ dt ρ dt ξ B T ρ dt T ρ Cρ ( ) Cρ / Ĩ ( T ) B ϕ dx dt x σ( ) ξ ξ dt dx B x σ( ) ξ Ĩ ( T ) = d T e dτ T dτ B ξ x σ( ) ξ d T e dτ dτ. T ξ x σ( ) ξ dx, dx (.) Snce for x = n the last ntegral (.) there s a sngularty, then we estmate t separately. B ρ = = ξ dx x σ( ) ξ ρ (µ r n λ µ r n λ r n λ ) r n r σ( ) (µ r n λ µ r n λ r n λ ) r λ λ ( ) σ( n ) n (µ µ r λ r λ λ ) dr. (µ µ r λ r λ λ ) dr (.5)

8 8 S. BAGIROV EJDE-8/9 Usng the L Hoptal s rule, we obtan µ µ lm r λ r λ λ r µ µ r λ r λ λ λ = lm µ r λ (λ r λ µ r λ (λ λ )r λ λ λ )r λ λ = λ D λ λ λ λ D λ λ λ Then there exsts r > such that for r < r, = D D. D < µ µ r λ r λ λ D µ µ r λ r λ λ < D D. So, for r < r, µ µ r λ r λ λ < ( D )( µ µ ) D r λ r λ λ. On the other hand, for r r, µ µ r λ r λ λ µ µ r λ r λ λ C(r ). Takng nto account the above two relatons, from (.5) we obtan where B ξ x σ( ) ξ dx C ρ ρ r λ λ ( ) σ( n ) dr (λ = C r λ n σ ( )) dr η ρ, for η > C ln(ρ), for η =, forη <, η = λ λ σ n ( ). Usng (.6), from (.) we obtan Ĩ ( T (η )ρ ), for η > I C ln(ρ)ρ /, for η = ρ /, for η <. To estmate J, =,, we estmate each addend of H (ξ, ϕ) separately. ( ( ξ ), ϕ) 3 ξ r 3 n ξ r r n ξ ϕ r r r Cr n 3 λ ϕ, r (.6) (.7)

9 EJDE-8/9 NONEXISTENCE OF GLOBAL SOLUTIONS 9 ξ ϕ ξ r n ξ ϕ r r r n ϕ r r Cr n λ ϕ r n ϕ, r r ( ξ ( ϕ)) Cr n λ 3 ϕ r 3 n ϕ r r n ϕ, r r n ξ ϕ x x x x,= n,= n,= ( ξ x x r r ) ( ϕ x r ξ x x r r ξ r (δ x r ) r x x r 3 ) ϕ x x r r ϕ (δ r r x x ) r 3 ( C ξ r )( r ξ r ϕ r ) r ϕ r Cr n ( λ ϕ r ) r ϕ r. Now, takng nto account these relatons and (.3), we estmate J, =, : J = ρ ρ H (ξ, ϕ) Bρ,ρ Tρ dx dt x σ( ) ξ ϕ T ρ dt H (ξ, ϕ) Bρ,ρ dx x σ( ) ξ ϕ n ( Cρ λ ) σ( n ) ( λ )( )n ( d3 eϕ ds d eϕ 3 ds d eϕ ds ) s σ( ) ϕ Cρ ( )λ λ ( ) σ( n ) J ( ϕ) (η = Cρ ) J ( ϕ), ds (.8) where J ( ϕ) denotes the last ntegral. Usng the estmates (.7),(.8), we estmate the rght-hand sdes of (.), (.). It s known that for large κ and γ, the ntegrals Ĩ( T ), J ( ϕ) are bounded [7]. Dependng on the sgn of η, =,, we consder varous varants. I. α >, α >. Ths s euvalent to λ λ σ > and λ λ σ >. (.9) Subect to relaton (.9), we consder the followng cases. (a) η, η or α, α. Then, takng nto account (.7), (.8), from (.), (.) we obtan x σ u T ρ ξ ϕ dx dt

10 S. BAGIROV EJDE-8/9 where Cρ ()[ f Ĩ / f (ρ) = J ] / [ f {, f η < ln(ρ), f η =. When we pass to lmt as ρ, we obtan x σ u T ρ ξ ϕ dx dt. Ĩ J ], Hence u, u. (b) Now let η >, η > or > α, > α. Agan usng (.7), (.8), from (.), (.) we obtan x σ u T ρ ξ ϕ dx dt Assume that Snce Cρ [ Ĩ ((η )η )[ I / ( T ) J ( T ) ]. ( T ) J / ( T ) ] (.) mn{ (η ) η, (η ) η } <. (.) (η ) η = λ λ σ n ( ) λ λ σ n ( ) = ( )θ, then we can wrte (.) as max(θ, θ ) >. For defnteness, we assume (η ) η <. Then for =, from (.) we obtan x σ u T ρ ξ ϕ dx dt Cρ ((η )η )[ Ĩ J / Passng to the lmt as ρ, we obtan x σ u ξ dx dt. ] [ Ĩ / J ] /. Hence u. Then from the second euaton of the system t follows that u. Smlarly, for (η ) η <, we obtan u, u. Now let mn{ (η ) η, (η ) η } = or the same max(θ, θ ) =. For example, take (η ) η =. Then from (.) t follows x σ u T ρ ξ dx dt C.

11 EJDE-8/9 NONEXISTENCE OF GLOBAL SOLUTIONS From the propertes of the ntegral, t follows that ρ Then from (.9), by (.) and (.3) we obtan x σ u T ρ ξ ϕ dx dt So, agan [( B ( B x σ u ξ dx dt, (.) B ρ,ρ x σ u ξ dx dt. (.3) B x σ u ξ dx dt) /I / /J ] x σ u / ξ dx dt) /[ / I ] J / Cρ ( (η )η ) [( ( B ρ,ρ x σ u ξ dx dt B ) / x σ u ξ dx dt) Ĩ / J / x σ u ξ dx dt. ] /[Ĩ/ J ] /. Hence u and respectvely u. If (η ) η =, then n the same way, we obtan u, u. (c) Let us consder the case when η, η. At frst, let η, η. As n the prevous cases, from (.), (.) we obtan x σ u ξ dx dt Cρ ( ()η)[ f / Ĩ / J ] / [ Ĩ / J ] /. If η < ( ), then passng to lmt as ρ, from (.) we have x σ u ξ dx dt. (.) Hence u and from the second euaton of the system t follows u. Note that f η <, then for η = ( ), f rom (.) we obtan x σ u ξ dx dt < C. As n the prevous case, we can show agan that u, u. Note that the condton η <, η ( ) s euvalent to the condton < < α, α β,

12 S. BAGIROV EJDE-8/9 and the condton η =, η ( ) to the condton = α, α < β. Now let η, η. Then smlar to the prevous case we obtan that for η <, η ( ) and for η =, η < ( ), u, u. The same condton η <, η ( ) s euvalent to the condton < α, α β, and the condton η =, η < ( ) to the condton = α, α < β. II. α, α >. Herewth, the cases η, η > and η >, η > should be consdered. For η, η > as n the prevous cases, we obtan u, u f η <, η ( ) and η =, η < ( ). From the neualty η ( ) t follows that < β. Snce β = λ σ n λ n, ths case has meanng for λ σ 8 > λ. Now let η >, η >. Then smlar to case (b), we obtan that u, u f > α, > α, max{θ, θ }. III. α >, α. Herewth, t s necessary to consder the case when η >, η and η >, η >. For η >, η, u, u f < α, < β and = α, < < β, and n the case η >, η >, for > α, < < β, max{θ, θ }. Obvously, ths case has meanng for β > or for λ σ 8 > λ. IV. α, α. Here t s necessary to consder the only case when η >, η >. Then u, u, f < < β, < < β and max{θ, θ }. Obvously, ths set s not empty f λ σ 8 > λ, λ σ 8 > λ. Ths completely proves the theorem. Note that remans open the cases = α, = β and = β, = α. References [] D. Andreucc, M. A. Herrero, J. J. L. Velazuez; Louvlle theorems and blow up behavor n semlnear reacton dffuson systems, Ann. Inst. Henr Poncarce, Anal. Non Lnneare, (997), -53. [] Sh. G. Bagyrov; On the exstence of a postve soluton of a nonlnear second-order parabolc euaton wth tme-perodc coeffcents, Dfferentsal nye Uravnenya, 7, Vol. 3, No., pp [3] M. F. Bdaut-Veron, S. Pohozaev; Nonexstence results and estmates for some nonlnear ellptc problems, Anal. Math., 8 (), -9. [] K. Deng, H. A. Levne; The role of crtcal exponents n blow-up theorems: the seuel, J. Math. Anal. Appl., 3 (), no., 856. [5] M. Escobedo, M. A. Herrero; Boundedness and blow up for a semlnear reactondffuson system, J. Dfferental Euatons, 89 (99), 76-.

13 EJDE-8/9 NONEXISTENCE OF GLOBAL SOLUTIONS 3 [6] M. Fla, A. Levne, Y. A. Uda; Futa-type global exstence-global nonexstence theorem for a system of reacton dffuson euatons wth dfferng dffusvtes, Math. Methods Appl. Sc., 7 (99), [7] H. Futa; On the blowng-up of solutons of the Cauchy problem for u t = u u α, J. Fac. Sc. Unv, Tokyo, Sect. I, 3 (966), 9-. [8] Gabrella Carst, Exstence and nonexstence of global solutons of degenerate and sngular parabolc system, Abstr. Appl. Anal., 5 (), no., 65-8 [9] B. Gdas, J. Spruck; Global and local behavor of postve solutons of lnear ellptc euatons, Comm. Pare. Appl. Math.vol 3,pages ,yr 98 [] K. Hayakawa; On nonexstence of global solutons of some sem-lnear parabolc euatons, Proc. Japan. Acad., 9 (973), [] K. Kobayash, T. Saro, H. Tanaka; On the blowng up problem of sem lnear heat euatons, J. Math. Soc. Japan, 9 (977), 7- [] A. A. Kon kov; On solutons of uas-lnear ellptc neualtes contanng terms wth lowerorder dervatves, Nonlnear Anal., 9 (3), pages -3. [3] G. G. Laptev; On nonexstence of solutons of a class of sngular semlnear dfferental neualtes, Tr. MIAN, 3 (), [] H. A. Levne; The role of crtcal exponents n blowup theorems, SIAM Revew, 3 (99), no., [5] H. A. Levne; A Futa type global exstence global nonexstence theorem for a weakly coupled system of reacton-dffuson euatons, Zet. Ang. Math.Phys., (99), 8-3. [6] E. Mtder, S. I. Pohozaev; Absence of postve solutons for uas-lnear ellptc problems on R N, Proc. Steklov Inst. Math., 7 (999), 86-6 (n Russan). [7] E. Mtder, S. I. Pohozaev; A pror estmatons and no solutons of nonlnear partal euatons and neualtes, Proc. of V. A. Steklov Mathematcs Insttute of NAS, 3 (), 9-3. [8] K. Mochzuk, Q. Huang; Exstence and behavor of solutons for a weakly coupled system of reacton-dffuson euatons, Methods Appl. Anal., 5 (998), 9-. [9] R. G. Pnsky; Exstence and nonexstence of global solutons for u t u = a(x)u n R d, Jour. Dff. Euatons, 33 (997), [] A. A. Samarsk, V. A. Galaktonov, S. P. Kurdyumov, A. P. Mkhaylov; Blowup of solutons n problems for uaslnear parabolc euatons, Nauka Pub., Moscow, 987. (In Russan). [] J. Serrn, H. Zou; Nonexstence of postve solutons of Lane-Emden system, Dff. Integr. Euat., 9 (996), pages [] Y. Uda; The crtcal exponent for a weakly coupled system of the generalzed Futa type reacton-dffuson euatons, Z. Angew. Math. Phys., 6 (995), no. 3, Shrmayl Bagrov Insttute of Mathematcs and Mechancs of NAS of Azerbaan, Baku, Azerbaan E-mal address: sh bagrov@yahoo.com