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The local well-posedness of solutions for a nonlinear pseudo-parabolic equation
Boundary Value Problems volume 2014, Article number: 177 (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.
The pseudo-parabolic equation possesses the form
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.
In this work, we study the following nonlinear pseudo-parabolic equation:
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.
Let () be the space of all measurable functions such that . We define with the standard norm . For any real number , denotes the Sobolev space with the norm defined by
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.
We consider the Cauchy problem for Eq. (2)
which is equivalent to
where is the inverse operator of .
Now, we give our main results for problem (3).
Letwith. Then the Cauchy problem (3) has a unique solutionwhereis the maximum existence time. Moreover,
if and only if
For the case of space dimension , we have the result.
3 Several lemmas
Letandbe real numbers such that. Then
(Kato and Ponce )
If, thenis an algebra. Moreover,
whereis a constant depending only on.
Assumewith. Then problem (3) admits a unique local solution
For the first equation of problem (4), we have
Letting functions and be in the closed ball of radius about the zero function in and letting be the operator on the right-hand side of (5), for fixed , we get
Using Lemma 3.1 derives
where and is independent of . Choosing sufficiently small such that , we know that operator is a contractive mapping. Applying the above inequality and (5) yields
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. □
Using and the Parseval equality gives rise to
For , applying on both sides of the first equation of system (3), noting the above equality and integrating the resultant equation with respect to by parts, we obtain the equation
For the terms and , we have
For the terms and , using Lemma 3.2 gives rise to
which results in (11). □
Proof of Theorem 2.1
Using Lemma 3.4, for any , we have
For , the Sobolev imbedding theorem yields
Proof of Theorem 2.2
For the space dimension , we write problem (3) in the form
Using for any integer and integration by parts, we have
which results in
from which we obtain
If is an odd integer, , and , we get
Using the conclusion of Theorem 2.1, we finish the proof of Theorem 2.2. □
Proof of Theorem 2.3
For and , using (22) yields
from which we obtain
which together with Theorem 2.1 completes the proof of Theorem 2.3. □
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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).
The authors declare that they have no competing interests.
The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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Lai, S., Yan, H. & Wang, Y. The local well-posedness of solutions for a nonlinear pseudo-parabolic equation. Bound Value Probl 2014, 177 (2014). https://doi.org/10.1186/s13661-014-0177-4
- local strong solution
- nonlinear pseudo-parabolic equation