Abstract
The Burgers-like equation is considered: \partial_ t u − Delta u + div G ( 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.
Highlights
We consider a viscous Burgers-like equation of the form (E)∂tu − ∆u + divG(u) = 0 in Rn × (0, T ), u|t=0 = u0 in Rn, where ∂t = ∂/∂t
In this paper we consider the case that u0 is not bounded at the space infinity
This paper specifies the growth of nonlinear term as G(r) ∼ r2 for large r
Summary
Theorem (Existence and uniqueness of a solution of a viscous Burgers like equation). (The transformation form v to w is called the Hopf-Cole transformation.) Our problem is reduced to the unique solvability of the heat equation with initial data w ∼ eax for large x. Like their result it is possible to prove the global existence when the growth order is less than linear We shall discuss this topic in a forthcoming paper of the second author. T uk+1(t) = et∆u0 − e(t−s)∆∇uk+1(s) G (uk(s))ds To use this iteration (2) it is necessary to study the solvability of the linear equation with growing coefficients in the transport term:. It is not very difficult to solve the linear equation (3) for initial data v0 ∈ BC, where BC is the set of all bounded continuous functions and BCm is defined by BCm =. Α must satisfy α ≤ 2α − 1 so that α ≤ 1
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