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

Gao and Gao Boundary Value Problems 2013, 2013:76 http://www.boundaryvalueproblems.com/content/2013/1/76 RESEARCH Open Access Study of solutions to an initial and boundary value problem for certain systems with variable exponents Yunzhu Gao1* and Wenjie Gao2* * Correspondence: ; 1 Department of Mathematics and Statistics, Beihua University, Jilin City, P.R. China 2 Institute of Mathematics, Jilin University, Changchun, 130012, P.R. China Abstract 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. Keywords: existence; blow-up; parabolic system; hyperbolic system; variable exponent 1 Introduction In this paper, we first consider the initial and boundary value problem to the following nonlinear parabolic system with variable exponents: ⎧ ⎪ ut = u + f (u, v), (x, t) ∈ QT , ⎪ ⎪ ⎪ ⎪ ⎨v = v + f (u, v), (x, t) ∈ Q , t  T ⎪ ⎪u(x, t) = , v(x, t) = , (x, t) ∈ ST , ⎪ ⎪ ⎪ ⎩ u(x, ) = u (x), v(x, ) = v (x), x ∈ , (.) where  ⊂ RN is a bounded domain with smooth boundary ∂ 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) , or   vp (y) (y, t) dy f (u, v) = a (x)  and up (y) (y, t) dy, f (u, v) = a (x)  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, © 2013 Gao and Gao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Gao and Gao Boundary Value Problems 2013, 2013:76 http://www.boundaryvalueproblems.com/content/2013/1/76 Page 2 of 10 boundedness and blow-up properties of the solutions; we refer the readers to the bibliography given in [–]. The motivation of this work is due to [], where the following system of equations is studied. ⎧ ⎨u – u = vp , t ⎩vt – v = uq , (.) 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: ⎧ ⎪ ⎪ ⎨ut = u + f (x, u), (x, t) ∈  × [, T), u(x, ) = u (x), x ∈ , ⎪ ⎪ ⎩ u(x, t) = , (x, t) ∈ ∂ × [, T), (.) 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: ⎧ ⎪ utt = u + f (u, v), (x, t) ∈ QT , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨vtt = v + f (u, v), (x, t) ∈ QT , u(x, t) = , v(x, t) = , (x, t) ∈ ST , ⎪ ⎪ ⎪ ⎪ v(x, ) = v (x), x ∈ , u(x, ) = u (x), ⎪ ⎪ ⎪ ⎪ ⎩ ut (x, ) = u (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 In this section, we first state some assumptions and definitions needed in the proof of our main result and then prove the existence of solutions. Gao and Gao Boundary Value Problems 2013, 2013:76 http://www.boundaryvalueproblems.com/content/2013/1/76 Page 3 of 10 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    (u, v) → ∞ as t → T ∗ , where        (u, v) = sup u(·, t) + v(·, t) . ∞ ∞ t∈[,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  . Proof We only prove the case when f (u, v) = a (x)vp (x) and f (u, v) = a (x)up (x) , and the   proofs to the cases f (u, v) = a (x)  vp (y) (y, t) dy and f (u, v) = a (x)  up (y) (y, t) dy are similar. Let us consider the equivalent systems of (.) ⎧ ⎨u(x, t) =  g(x, y, t)u (y) dy +  t  g(x, y, t – s)a (y)vp (y) dy ds,      ⎩v(x, t) =  g(x, y, t)v (y) dy +  t  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) =  t g(x, y, t)u (y) dy +     t  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  g(x, y, t – s)a (y)vp (y) dy ds, n     ⎩ (u) =  t  g(x, y, t – s)a (y)upn (y) dy ds   is a contraction in the set ET to be defined below. Gao and Gao Boundary Value Problems 2013, 2013:76 http://www.boundaryvalueproblems.com/content/2013/1/7 (...truncated)