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A nonlinear oblique derivative boundary value problem for the heat equation. I: Basic results. (English) Zbl 0922.35072

The authors examine the following boundary value problem from plasma physics: \[ \begin{gathered} B_t - B_{XX} -B_{ZZ} =0\quad \text{ for }t>0,\;X>0,\;Z\in \mathbb R, \\ B_X-KBB_Z =0 \quad\text{ for } t>0,\;X=0,\;Z\in \mathbb R, \\ B(t,-\infty,X)=1,\;B(t,\infty,X)=0 \quad\text{for } t>0,\;X> 0, \end{gathered} \] where \(K\) is a given positive number. The interest in this paper is in self-similar solutions of the problem. Using the variables \(x=Z/(t+1)^{1/2}\), \(x=X/(t+1)^{1/2}\), these are functions \(U(x,z)\) such that \[ \begin{gathered} -U_{xx}- \tfrac 12 (zU_z +xU^x) =0 \quad\text{ for } x>0,\;z\in \mathbb R, \\ U_x-KUU_z=0 \quad\text{ for } x=0, z\in \mathbb R, \\ U(-\infty,x)=1,\;U(\infty,x)=0 \quad\text{ for } x>0. \end{gathered} \] By a series of a priori estimates, the authors show that there is a unique self-similar solution and then that the solution \(B\) of the original problem converges exponentially fast to this solution (as \(t\to\infty\)) provided \(B(0,z,x)\) converges to \(U(x,z)\) fast enough as \(x+| z| \to\infty\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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