## 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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### References:

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