A pseudoparabolic equation with nonlocal pu(x,t)$p\left[u(x,t)\right]$ ‐ Laplace operator

Khonatbek Khompysh and Sergey Shmarev

Mathematische Nachrichten, 2025, vol. 298, issue 12, 3832-3854

Abstract: We study the Dirichlet problem for the pseudoparabolic equation perturbed with the p[u]$p[u]$‐Laplacian diffusion term, ut−γΔut−βdiv∇up[u]−2∇u=f(x,t),$$\begin{equation*} \hspace*{6pc}{u}_t - \gamma \Delta {u}_t -\beta \operatorname{div}{\left({\left|\nabla {u}\right|}^{p[{u}]-2} \nabla {u}\right)} ={f}({x},t), \end{equation*}$$where the argument of the exponent p[u]$p[{u}]$ depends on the sought solution: p[u]≡p(l(u)),l(u)=∫Ωg(x,u(x,t))dx$$\begin{equation*} \hspace*{6.5pc}p[{u}]\equiv p(l({u})), \quad l(u)=\int \limits _\Omega g(x,u(x,t))\, d{x} \end{equation*}$$with a given function g(·,·)$g(\cdot,\cdot)$. Under suitable conditions on the problem data, we prove the existence and uniqueness of a weak solution. It is shown that the solutions of the pseudoparabolic problem converge to the solution of the corresponding parabolic problem as γ→0$\gamma \rightarrow 0$.

Date: 2025
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