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On the first boundary value problem for quasilinear parabolic equations with two independent variables. (English) Zbl 0962.35094

Let \(l\) and \(T\) be positive numbers and set \(Q_T=(-l,l)\times (0,T)\). The parabolic differential equation \[ -u_t + a(X,u,u_x)u_{xx} =f(X,u,u_x)\quad \text{in } Q_T \] with boundary conditions \(u(\pm l,t)=0\) for \(t>0\) and initial condition \(u(\cdot,0) =u_0\) has been studied (as part of a much more general theory) for a long time. S. N. Kruzhkov [Trudy Semin. Im. Petrovskogo 5, 217-272 (1979; Zbl 0434.35058)] showed that this problem has a solution if \(a\) and \(f\) are sufficiently smooth (for example, Hölder continuous with respect to all variables) and \(a\) and \(f\) satisfy the inequality \[ |f(X,z,p)|\leq a(X,z,p)\psi(|p|) \] (so, in particular \(a >0\)) for some function \(\psi\) such that the integral \(\int^\infty (\rho/\psi(\rho)) d\rho=\infty\). When this integral is finite, it is not hard to show that this initial-boundary value problem need not have a solution. The purpose of this paper is to refine this result by finding restrictions on \(u_0\) and the oscillation of \(u\) in terms of this integral which imply existence of a solution. The key step is a boundary gradient estimate which is proved by a more careful consideration of the usual method for proving such an estimate.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B45 A priori estimates in context of PDEs

Citations:

Zbl 0434.35058
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References:

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