Study of solutions to an initial and boundary value problem for certain systems with variable exponents

Boundary Value Problems, Apr 2013

In this paper, the existence and blow-up property of solutions to an initial and boundary value problem for a nonlinear parabolic system with variable exponents is studied. Meanwhile, the blow-up property of solutions for a nonlinear hyperbolic system is also obtained.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.boundaryvalueproblems.com/content/pdf/1687-2770-2013-76.pdf

Study of solutions to an initial and boundary value problem for certain systems with variable exponents

Boundary Value Problems Study of solutions to an initial and boundary value problem for certain systems with Yunzhu Gao 1 2 3 Wenjie Gao 0 1 2 0 Institute of Mathematics , Jilin 1 City , P.R. China 2 Statistics, Beihua University , Jilin 3 Department of Mathematics system is also obtained. In this paper, the existence and blow-up property of solutions to an initial and boundary value problem for a nonlinear parabolic system with variable exponents is studied. Meanwhile, the blow-up property of solutions for a nonlinear hyperbolic exponent; existence; blow-up; parabolic system; hyperbolic system; variable 1 Introduction In this paper, we first consider the initial and boundary value problem to the following nonlinear parabolic system with variable exponents: u + f(u, v), (x, t) ∈ QT , v + f(u, v), (x, t) ∈ QT , v(x, t) = , (x, t) ∈ ST , v(x, ) = v(x), x ∈ , and  < T < ∞, QT = × [, T ), ST denotes the lateral boundary of the cylinder QT , and the source terms f, f are in the form f(u, v) = a(x)vp(x) and f(u, v) = a(x)up(x), f(u, v) = a(x) vp(y)(y, t) dy and f(u, v) = a(x) up(y)(y, t) dy, respectively, where p, p, a, a are functions satisfying conditions (.) below. In the case when p, p are constants, system (.) provides a simple example of a reaction-diffusion system. It can be used as a model to describe heat propagation in a two-component combustible mixture. There have been many results about the existence, u = vp, v = uq, u + f (x, u), (x, t) ∈ × [, T ), × [, T ), where x ∈ RN (N ≥ ), t > , and p, q are positive numbers. The authors investigated the boundedness and blow-up of solutions to problem (.). Furthermore, the authors also studied the uniqueness and global existence of solutions (see []). Besides, this work is also motivated by [] in which the following problem is considered: where ∈ Rn is a bounded domain with smooth boundary ∂ , and the source term is of the form f (x, u) = a(x)up(x) or f (x, u) = a(x) uq(y)(y, t) dy. The author studied the blowup property of solutions for parabolic and hyperbolic problems. Parabolic problems with sources like the ones in (.) appear in several branches of applied mathematics, which can be used to model chemical reactions, heat transfer or population dynamics etc. We also refer the interested reader to [–] and the references therein. We also study the following nonlinear hyperbolic system of equations: v(x, t) = , (x, t) ∈ ST , ⎪⎪⎪⎪⎩⎪⎪⎪⎪ uut((xx,,))==uu((xx)),, v(x, ) = v(x), x ∈ , vt(x, ) = v(x), x ∈ . The aim of this paper is to extend the results in [, ] to the case of parabolic system (.) and hyperbolic system (.). As far as we know, this seems to be the first paper, where the blow-up phenomenon is studied with variable exponents for the initial and boundary value problem to some parabolic and hyperbolic systems. The main method of the proof is similar to that in [, ]. We conclude this introduction by describing the outline of this paper. Some preliminary results, including existence of solutions to problem (.), are gathered in Section . The blow-up property of solutions are stated and proved in Section . Finally, in Section , we prove the blow-up property of solutions for hyperbolic problem (.). 2 Existence of solutions Throughout the paper, we assume that the exponents p(x), p(x) : → (, +∞) and the continuous functions a(x), a(x) : → R satisfy the following conditions:  < p– = inf p(x) ≤ p(x) ≤ p+ = sup p(x) < +∞, x∈ x∈  < p– = inf p(x) ≤ p(x) ≤ p+ = sup p(x) < +∞, x∈ x∈  < c ≤ a(x) ≤ C < +∞,  < c ≤ a(x) ≤ C < +∞. Definition . We say that the solution (u(x, t), v(x, t)) for problem (.) blows up in finite time if there exists an instant T ∗ < ∞ such that as t → T ∗, u(·, t) ∞ + v(·, t) ∞ . Our first result here is the following. Theorem . Let ⊂ RN be a bounded smooth domain, p(x), p(x), a(x), a(x) satisfy the conditions in (.), and assume that u(x) and v(x) are nonnegative, continuous and bounded. Then there exists a T ,  < T  ≤ ∞, such that problem (.) has a nonnegative and bounded solution (u, v) in QT . g(x, y, t – s)a(y)vp(y) dy ds, g(x, y, t – s)a(y)up(y) dy ds, where g(x, y, t) is the corresponding Green function. Then the existence and uniqueness of solutions for a given (u(x), v(x)) could be obtained by a fixed point argument. We introduce the following iteration scheme: u(x, t) = , v(x, t) = , un+(x, t) = g(x, y, t)u(y) dy + vn+(x, t) = g(x, y, t)v(y) dy + g(x, y, t – s)a(y)vpn(y) dy ds, g(x, y, t – s)a(y)upn(y) dy ds, and the convergence of the sequence {(un, vn)} follows by showing that (v) = t (u) = t g(x, y, t – s)a(y)vpn(y) dy ds, g(x, y, t – s)a(y)upn(y) dy ds is a contraction in the set ET to be defined below. Now, we define (v) = (u) = g(x, y, t – s)vpn(y) dy ds, g(x, y, t – s)upn(y) dy ds. (u, v) – (w, z) = (v) – (z), (u) – (w) , and for arbitrary T > , define the set ET = C,( T ) ∩ C( T ) (u, v) ≤ M , where T = × [, T ], M > |(u(x), v(x)) | is a fixed positive constant. We claim that is a contraction on ET . In fact, for any x ∈ fixed, we have and we always have Now, we define It is obvious that μ(t) →  when t → +. Then, by using inequality (.), we get g(x, y, t – s) dy ds. pi(x)wipi(x)–(ξi – ηi) ≤ pi+(M)pi+– ξi – ηi ∞, i = , . (u) – (w) ∞ a(y)g(x, y, t – s) vpn(y) – znp(y) dy ds a(y)g(x, y, t – s) upn(y) – wpn(y) dy ds  ≤ μ(t) (M)max{p+,p+}–(C + C) p ∞ + p ∞ ≤ μ(t) (M)max{p+,p+}–(C + C) p ∞ + p ∞ = μ(t) (M)max{p+,p+}–(C + C) p ∞ + p ∞ (v – z, u – w) . v – z ∞ + u – w ∞ Hence, for sufficiently small t, we have (u, v) – (w, z) (v) – (z), (u) – (w) (v) – (z) ∞ + (u) – (w) ∞ ≤ μ(t) (M)max{p+p+}–(C + C) p ∞ + p ∞ (v – z, u – w) is a strict contraction. 3 Blow-up of solutions In this section, we study the blow-up property of the solutions to problem (.). We need the following lemma. Lemma . Let y(t) be a solution of Proof It is sufficient to take K >  such that Hence, we have y (t) ≥ a y(t)r. where λ > , a > , r >  and C >  are given constants. Then, there exists a constant K >  such that if y() ≥ K , then y(t) cannot be globally defined; in fact, By a direct integration to (.), then we get immediately (.), which gives an upper bound for the blow-up time t∗ = ay(r––r()) . The next theorem gives the main result of this section. Theorem . Let ⊂ RN be a bounded smooth domain, and let (u, v) be a positive solution of problem (.), with p(x), p(x), a(x), a(x) satisfying conditions in (.). Then any solutions of problem (.) will blow up at finite time T ∗ if the initial datum (u(x), v(x)) satisfies u(x) + v(x) ϕ > C, where ϕ >  is the first eigenfunction of the homogeneous Dirichlet Laplacian on and C >  is a constant depending only on the domain and the bounds C, C given in condition (.). x ∈ ϕ dx = . We introduce the function η(t) = a(x)vp(x), f(u, v) = a(x)up(x). Then (ut + vt)ϕ with the homogeneous Dirichlet boundary condition, and let ϕ be a positive function satisfying a(x)vp(x) + a(x)up(x) ϕ a(x)vp(x) + a(x)up(x) ϕ. We now deal with the term the following four sets: (a(x)vp(x) + a(x)up(x))ϕ. For each t > , we divide  = x ∈  = x ∈ : v(x, t) < , u(x, t) <  ,  = x ∈ : v(x, t) < , u(x, t) ≥  , : v(x, t) ≥ , u(x, t) <  ,  = x ∈ : v(x, t) ≥ , u(x, t) ≥  . Then we have a(x)vp(x) + a(x)up(x) ϕ a(x)vp(x) + a(x)up(x) ϕ + a(x)vp(x) + a(x)up(x) ϕ a(x) vp(x) + a(x)up(x) ϕ + a(x)vp(x) + a(x)up(x) ϕ a(x)upϕ + a(x)vpϕ + a(x)vp + a(x)up ϕ a(x)upϕ + a(x)vp + a(x)up ϕ – a(x)vp + a(x)up ϕ a(x)vp + a(x)up ϕ – a(x)vp + a(x)up ϕ a(x)vp(x) + a(x)up(x) ϕ ≥ γ Then we get η (t) ≥ –λη(t) + γp– ηp(t) –  . (u + v)ϕ ≤ u(·, t) L∞( ) + v(·, t) L∞( ) ϕ ≤ (u, v) . Hence, for η() big enough, the result follows from Lemma .. Next, we state briefly the proof to the theorem in the case f(u, v) = a(x) vp(y)(y, t) dy and f(u, v) = a(x) up(y)(y, t) dy. We repeat the previous argument under defining η(t) = (u + v)ϕ, and we obtain in much the same way vp(y)(y, t) dy + a(x) up(y)(y, t) dy ϕ(x) dx. In view of the property of ϕ, we get vp(y)(y, t) dy + a(x) up(y)(y, t) dy ϕ(x) dx vp(y)(y, t) dy a(x)ϕ(x) dx + up(y)(y, t) dy a(x)ϕ(x) dx ≥ c vp(y)(y, t) + up(y)(y, t) According to the convex property of the function f (w) = wr, r > , and by using Jensen’s inequality, by considering again , , ,  as before, we obtain vp(y)(y, t) + up(y)(y, t) η (t) ≥ –λη(t) + γηp(t) –  | |. By Lemma ., the proof is complete. 4 Blow-up of solutions for a hyperbolic system Lemma . [] Let y(t) ∈ C satisfying y (t) ≥ h y(t) , t ≤ Now, let us study the following problem: ⎨⎪⎪⎪⎪⎪⎪⎪⎧⎪ vuutt(ttx==,t) =vu++,ff((uu,,vvv))(,,x, t()(xx=,,tt)),∈∈QQ(xTT,,,t) ∈ ST , ⎪⎪⎪⎪⎩⎪⎪⎪⎪ uut((xx,,))==uu((xx)),, vv(tx(,x,))==vv(x(x),), xx∈∈ , , where u(x), v(x), u(x), v(x) ≥  and they are not identically zero, and f(u, v), f(u, v) as above respectively. Theorem . Let (u, v) ∈ C × C be a solution of problem (.), and let the conditions in (.) hold. Then there exist sufficiently large initial data u, v, u, v such that any solutions of problem (.) blew up at finite time T ∗. Proof Let (λ, ϕ) be the first eigenvalue and eigenfunction of Laplacian in with homogeneous Dirichlet boundary conditions as before. We assume that f(u, v) = a(x)vp(x), f(u, v) = a(x)up(x), the other is similar. We also define the function η(t) = (u + v)ϕ, so we have (utt + vtt)ϕ a(x)vp(x) + a(x)up(x) ϕ a(x)vp(x) + a(x)up(x) ϕ. (a(x)vp(x) + a(x)up(x))ϕ is dealt with as before, then we get a(x)vp(x) + a(x)up(x) ϕ ≥ γ ϕ ≥ we still obtain Then we have > , and note that (u + v)ϕ, (u + v)ϕ. (u + v)ϕ ≤ u(·, t) L∞( ) + v(·, t) L∞( ) ϕ ≤ (u, v) . Hence, (u, v) blows up before the maximal time of existence defined in inequality (.) is Competing interests The authors declare that they have no competing interests. Authors’ contributions YG performed the calculations and drafted the manuscript. WG supervised and participated in the design of the study and modified the draft versions. All authors read and approved the final manuscript. Acknowledgements Supported by NSFC (11271154) and by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University. 1. Chen , Y , Levine , S, Rao, M: Variable exponent, linear growth functions in image restoration . SIAM J. Appl. Math. 66 , 1383 - 1406 ( 2006 ) 2. Escobedo , M, Herrero, MA: Boundedness and blow up for a semilinear reaction-diffusion system . J. Differ. Equ . 89 , 176 - 202 ( 1991 ) 3. Escobedo , M, Herrero, MA: A semilinear parabolic system in a bounded domain . Ann. Mat. Pura Appl. CLXV , 315 - 336 ( 1998 ) 4. Friedman , A, Giga , Y: A single point blow up for solutions of nonlinear parabolic systems . J. Fac. Sci. Univ. Tokyo Sect. I 34 ( 1 ), 65 - 79 ( 1987 ) 5. Galaktionov , VA, Kurdyumov, SP, Samarskii, AA: A parabolic system of quasiliner equations I. Differ. Equ . 19 (12), 2133 - 2143 ( 1983 ) 6. Galaktionov , VA, Vázquez, JL: A Stability Technique for Evolution Partial Differential Equations . Progress in Nonlinear Differential Equations and Their Applications , vol. 56 . Birkhäuser, Boston ( 2004 ) 7. Kufner , A, Oldrich, J, Fucik, S: Function Space . Kluwer Academic, Dordrecht ( 1977 ) 8. Pinasco , JP: Blow-up for parabolic and hyperbolic problems with variable exponents . Nonlinear Anal . 71 , 1094 - 1099 ( 2009 ) 9. Andreu-Vaillo , F, Caselles , V, Mazón, JM : Parabolic Quasilinear Equations Minimizing Linear Growth Functions . Progress in Mathematics, vol. 223 . Birkhäuser, Basel ( 2004 ) 10. Antontsev , SN, Shmarev, SI : Anisotropic parabolic equations with variable nonlinearity . CMAF , University of Lisbon, Portugal 013 , 1 - 34 ( 2007 ) 11. Antontsev , SN, Shmarev, SI: Blow-up of solutions to parabolic equations with nonstandard growth conditions . CMAF , University of Lisbon, Portugal 02 , 1 - 16 ( 2009 ) 12. Antontsev , SN, Shmarev, SI : Parabolic equations with anisotropic nonstandard growth conditions . Int. Ser. Numer. Math. 154 , 33 - 44 ( 2007 ) 13. Antontsev , SN, Shmarev , S : Blow-up of solutions to parabolic equations with nonstandard growth conditions . J. Comput. Appl. Math. 234 , 2633 - 2645 ( 2010 ) 14. Erdem , D: Blow-up of solutions to quasilinear parabolic equations . Appl. Math. Lett . 12 , 65 - 69 ( 1999 ) 15. Glassey , RT: Blow-up theorems for nonlinear wave equations . Math. Z. 132 , 183 - 203 ( 1973 ) 16. Kalashnikov , AS: Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations . Russ. Math. Surv. 42 ( 2 ), 169 - 222 ( 1987 ) 17. Levine , HA: Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = - Au + F(u). Arch. Ration. Mech. Anal . 51 , 371 - 386 ( 1973 ) 18. Levine , HA, Payne, LE : Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations . J. Math. Anal. Appl . 55 , 329 - 334 ( 1976 ) 19. Lian , SZ, Gao, WJ, Cao, CL, Yuan, HJ: Study of the solutions to a model porous medium equation with variable exponents of nonlinearity . J. Math. Anal. Appl . 342 , 27 - 38 ( 2008 ) 20. Ruzicka , M: Electrorheological Fluids: Modelling and Mathematical Theory. Lecture Notes in Math. , vol. 1748 . Springer, Berlin ( 2000 ) 21. Simon , J: Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. 4(146) , 65 - 96 ( 1987 ) 22. Tsutsumi , M: Existence and nonexistence of global solutions for nonlinear parabolic equations . Publ. Res. Inst. Math. Sci . 8 , 211 - 229 ( 1972 ) 23. Zhao , JN: Existence and nonexistence of solutions for ut = div(|∇u|p- 2 ∇u) + f (∇u, u, x, t). J. Math. Anal. Appl . 172 , 130 - 146 ( 1993 )


This is a preview of a remote PDF: http://www.boundaryvalueproblems.com/content/pdf/1687-2770-2013-76.pdf

Yunzhu Gao, Wenjie Gao. Study of solutions to an initial and boundary value problem for certain systems with variable exponents, Boundary Value Problems, 2013, 76,