Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms

Abstract

In this paper we study the asymptotic behaviour of solutions of certain nonlinear parabolic equations with variable viscosity and geometric terms. We generalize the results on the large time behaviour and vanishing viscosity limits obtained earlier for planar Burgers equation by Hopf [7] Lighthill [20] and others. For several classes of systems of equations we derive explicit solution for initial value problem with different types of initial conditions and study large time behaviour of the solutions and its asymptotic form. We derive the simple hump solutions and $N$-wave solutions as its asymptotes depending on the conditions on the data and derive $L^p$ decay estimates for solutions and show that they depend on the variable viscosity coefficient and geometric terms. We also analyse the small viscosity limit of these solutions.

Keywords

• Parabolic equations
• exact solutions
• asymptotic behaviour