The local well-posedness of solutions for a nonlinear pseudo-parabolic equation
© Lai et al.; licensee Springer 2014
Received: 4 April 2014
Accepted: 1 July 2014
Published: 24 September 2014
The local existence and uniqueness of solutions for a nonlinear pseudo-parabolic equation are established in the Sobolev space with . In addition, we prove the global existence of solutions for two special cases of the equation.
MSC: 35Q35, 35Q51.
where constant , , and . If , Eq. (1) becomes the heat equation with sources. If , we call Eq. (1) as the pseudo-parabolic model (see Ting , Showalter and Ting ). The pseudo-parabolic equation has many important physical backgrounds such as the seepage of homogeneous fluids through a fissured rock , the unidirectional propagation of nonlinear dispersive long waves ,  and the aggregation of populations  (where is the population density). Equation (1) is employed in the analysis of nonstationary processes in the area of semiconductors , , where the term is regarded as the free electron density rate, term is regarded as the linear dissipation of the free charge current and is a source of free electron current. Equation (1) is also named a Sobolev type model or a Sobolev-Galpern type model .
The initial-boundary value problem and the initial problem for the linear pseudo-parabolic equation were investigated in , ,  where the existence and uniqueness of solutions for the equation were established. Various dynamic properties of solutions for nonlinear pseudo-parabolic equations, including singular pseudo-parabolic equations and degenerate pseudo-parabolic equations can be found in –. It is worth to mention that Kaikina et al. considered the superlinear case of the Cauchy problem for Eq. (1) with and showed the existence and uniqueness of the solutions. Furthermore, it was shown that the Cauchy problem for Eq. (1) has a unique global solution under the assumptions and sufficiently small initial value . The existence, uniqueness, and comparison principle for mild solutions of Eq. (1) were established in Cao et al. by whom the large time behavior of the solutions and the critical global existence exponent and the critical Fujita exponent for Eq. (1) were obtained.
where is an integer, and are constants, is a polynomial with order , , and . When , Eq. (2) reduces to Eq. (1). The existence and uniqueness of local solutions for Eq. (2) are established in the Sobolev space with . We find that the local solution in the space blows up if and only if . For the space dimension , assuming that the initial value , , and is an odd number, we find the global existence of solutions for Eq. (2). For the other case , , and initial value , we also acquire the global existence result of solutions for Eq. (2).
The rest of this paper is organized as follows. The main results are stated in Section 2. Several lemmas and the proofs of main results are given in Section 3.
2 Main results
Firstly, we state some notations.
For and nonnegative number , denotes the Frechet space of all continuous -valued functions on . We set and . For simplicity, throughout this article, we let denote any positive constant.
where is the inverse operator of .
Now, we give our main results for problem (3).
For the case of space dimension , we have the result.
3 Several lemmas
(Kato and Ponce )
whereis a constant depending only on.
Choosing sufficiently small such that , we know that maps to itself. It follows from the contractive mapping principle that the mapping has a unique fixed point in . This completes the proof. □
which results in (11). □
Proof of Theorem 2.1
Proof of Theorem 2.2
Using the conclusion of Theorem 2.1, we finish the proof of Theorem 2.2. □
Proof of Theorem 2.3
which together with Theorem 2.1 completes the proof of Theorem 2.3. □
This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
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