Existence of Global Solutions for Semilinear Pseudoparabolic Equations with Unbounded Coefficient

Abstract

In this work, we are interested in sign-changing solutions of the Cauchy problem $\\partial_tu-\\triangle\\partial_tu=\\alpha\\triangle u+V(x) |^\\sigma u$ in $\\mathbb{R}^n\\times(0,\\infty)$, $u|_{t=0}=u_0$, where $\\alpha,\\sigma>0$ are constants and $u_0,V$ are given functions. We put a rather mild assumption on the coefficient $V$ that it satisfies $ (x)|\\lesssim |^a$ as $ |\\to\\infty$ for a constant $a\\geq0$. Thus, in particular, it can be bounded. The function spaces considered are weighted Lebesgue spaces with a polynomial weight of order $b$, denoted by $L^{q,b}(\\mathbb{R}^n)$. After proving the boundedness of relevant operators, especially, the Bessel potential and the Green operators, we can establish the local existence of solutions for the Cauchy problem. Then, employing a modified interpolation estimate on the weight Lebesgue spaces, we can also prove the global existence of solutions provided the initial function $u_0$ is sufficiently small.