Abstract

The purpose of this article is to prove the existence and uniqueness of weak solutions for nonlinear parabolic problem whose model is \begin{align*} \begin{cases} \frac{\partial v}{\partial t}-\operatorname{div}\left[|\nabla v-\Theta(v)|^{q(x)-2}(\nabla v-\Theta(v))\right]+\beta(v)=f & \text { in }\quad Q_{T}:=(0, T) \times \Omega ,\\ v=0 & \text { on }\quad \Sigma_{T}:=(0, T) \times \partial \Omega, \\ v(\cdot, 0)=v_{0} & \text { in }\quad \Omega. \end{cases} \end{align*} We transform the parabolic problem into the elliptic problem by using time discretization technique by Euler forward scheme and Rothe method combined with the theory of variable exponent Sobolev spaces.

Keywords: Nonlinear parabolic problem; existence; weak solution; variable exponent; semi-discretization; uniqueness; Rothe method.

MSC: 35K55, 35A01, 35A02

DOI: 10.57016/MV-SBdK8647

Pages:  1--12