On viscous Burgers-like equations with linearly growing initial data - doi: 10.5269/bspm.v20i1-2.7521

Abstract

The Burgers-like equation is considered:

\partial_t u − Delta u + divG(u) = 0 in R^n \times (0,T).

In this paper we consider the case that the initial data  is not bounded at the space infinity. This paper specifies the growth of nonlinear term as G(r) ~ r^2 for large r. A typical example is the viscous Burgers equation. Our goal is to solve the initial value problem when the initial data may grow linearly at the space infinity. We shall prove that the problem admits a unique local regular solution.