Solutions of Nonlinear Parabolic Equations without Growth Restrictions on the Data
Date
2001-09-12
Authors
Boccardo, Lucio
Gallouet, Thierry
Vazquez, Juan Luis
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
The purpose of this paper is to prove the existence of solutions for certain types of nonlinear parabolic partial differential equations posed in the whole space, when the data are assumed to be merely locally integrable functions, without any control of their behaviour at infinity. A simple representative example of such an equation is
ut - ∆u + |u|s-1 u = f,
which admits a unique globally defined weak solution u(x, t) if the initial function u(x, 0) is a locally integrable function in ℝN, N ≥ 1, and the second member ƒ is a locally integrable function of x ∈ ℝN and t ∈ [0, T] whenever the exponent s is larger than 1. The results extend to parabolic equations results obtained by Brezis and by the authors for elliptical equations. They have no equivalent for linear or sub-linear zero-order nonlinearities.
Description
Keywords
Nonlinear parabolic equations, Global existence, Growth conditions
Citation
Boccardo, L., Gallouet, T., & Vazquez, J. L. (2001). Solutions of nonlinear parabolic equations without growth restrictions on the data. <i>Electronic Journal of Differential Equations, 2001</i>(60), pp. 1-20.