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Existence and Long-Time Behavior of Variational Solutions to a Class of Nonclassical Diffusion Equations in Noncylindrical Domains

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Abstract

We prove the existence and uniqueness of variational solutions to the following non-autonomous nonclassical diffusion equation

$$u_{t} - {\Delta} u_{t} - {\Delta} u + f(u)= g(x,t) $$

in a noncylindrical domain with the homogeneous Dirichlet boundary condition, under assumptions that the spatial domains are bounded and increase with time, the nonlinearity f satisfies growth and dissipativity conditions of Sobolev type, and the external force g is time-dependent. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor \(\hat {\mathcal {A}}=\{A(t): t\in \mathbb {R}\}\).

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Acknowledgments

The author would like to thank Cung The Anh for his suggestion and stimulating discussion on the subject of the paper. This work is supported by the Haiphong University.

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  1. Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam

    Nguyen Duong Toan

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  1. Nguyen Duong Toan
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Correspondence to Nguyen Duong Toan.

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Toan, N.D. Existence and Long-Time Behavior of Variational Solutions to a Class of Nonclassical Diffusion Equations in Noncylindrical Domains. Acta Math Vietnam 41, 37–53 (2016). https://doi.org/10.1007/s40306-015-0120-5

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