On a class of nonlinear nonclassical parabolic equations

Abstract

We consider a class of problems in which traces of the first spacial derivative of the solution appear in the coefficients of a parabolic partial differential equation. Such problems arise in free boundary problems and the determination of unknown coefficients in various parabolic differential equations. Existence, uniqueness and continuous dependence on the data are demonstrated. The analysis involves a priori estimates and the Schauder fixed point theorem. Overall, the article generalizes some of the results of Cannon and Yin (1989, Journal of Differential Equations, 79, 266–288).

Keywords:

• 35Kxx Parabolic partial differential equations
• 35R30 Inverse problems
• 35R35 free boundary value problems

Keywords:

• AMS Classifications: 35-XX partial differential equation