Solutions to some nonlinear parabolic equations in pseudomeasure spaces

Abstract

AbstractIn this paper we study the Cauchy problem in pseudomeasure spaces for some nonlinear parabolic equations including nonlinear heat equations, nonlinear hyper‐dissipative equations, generalized nonlinear Hamilton–Jacobi equations, nonlinear hyper‐dissipative equations with convective term and the Ginzburg–Landau equation with third‐order nonlinear growth. We prove the existence, uniqueness and asymptotic stability of the global‐in‐time solutions for small initial data in some pseudomeasure spaces. As a result, the existence and uniqueness of selfsimilar solutions to the equations are obtained. Furthermore, with the self‐similar solutions of the equations, the long time behavior of solutions to the problem are also discussed. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)