Integral Solution for a Parabolic Equation Driven by the p(x)-Laplacian Operator with Nonlinear Boundary Conditions and L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}$$\end{document} Data

We study the solvability of a class of parabolic equations having p(x)-growth structure and involving nonlinear boundary conditions with irregular data. We tackle our problem in a suitable functional setting by considering the so-called Lebesgue and Sobolev spaces with variable exponents. We establish two existence and uniqueness results of solutions to the studied problem. In the first one, we use a subdifferential approach to investigate the existence and uniqueness of a weak solution when the data are regular enough. Second, we assume that the data belong only to the space L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} and we show the existence and uniqueness of an integral solution to the considered model. Our main idea relies essentially on the use of accretive operator theory and involves some new techniques.