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Abstract
In this paper, we consider the biparabolic problem under nonlocal conditions with both linear and nonlinear source terms. We derive the regularity property of the mild solution for the linear source term while we apply the Banach fixed-point theorem to study the existence and uniqueness of the mild solution for the nonlinear source term. In both cases, we show that the mild solution of our problem converges to the solution of an initial value problem as the parameter epsilon tends to zero. The novelty in our study can be considered as one of the first results on biparabolic equations with nonlocal conditions.
Highlights
Let be a bounded domain in RN (N ≥ 1) with sufficiently smooth boundary ∂
We prove that the solution of problem (1.1)–(1.2) converges to a mild solution with the initial value problem for (1.1)
5 Conclusion In this paper, we considered a biparabolic equation under temporal nonlocal conditions with linear and nonlinear source terms
Summary
Let be a bounded domain in RN (N ≥ 1) with sufficiently smooth boundary ∂. Bulavatsky [7] studied some boundary value problems for biparabolic equations with nonlocal boundary conditions. Besma et al [5] considered the problem of approximating a solution of an ill-posed biparabolic problem in the abstract Hilbert space. They introduced a modified quasi-boundary value method to get stable solutions for regularizing the ill-posedness of a biparabolic equation. Tuan et al [32] studied the problem of finding the initial distribution for a linear inhomogeneous or nonlinear biparabolic equation. To the best of our knowledge, up to date, there is still no any study considering problem (1.1) under the nonlocal condition (1.2) This motivates us to focus on problems (1.1)– (1.2).
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