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A nonlinear oblique derivative boundary value problem for the heat equation. II: Singular self-similar solutions. (English) Zbl 0945.35047
The authors continue their study of the following boundary value problem from plasma physics: \[ \begin{alignedat}{2} B_t - B_{XX} -B_{ZZ} &=0 && \text{ for }t>0,\;X>0,\;Z\in \mathbb R, \\ B_X-KBB_Z &=0 && \text{ for } t>0,\;X=0,\;Z\in \mathbb R, \\ B(t,-\infty,X)=1,\;B(t,\infty,X) &=0 &&\text{ for } t>0,\;X> 0, \\ B(0,Z,X) &=B^0(X,Z)\quad && \text{ for } X>0, Z\in \mathbb R, \end{alignedat} \] where \(K\) is a given positive number. In part 1 [ibid. xx, 221-253 (1999; Zbl 0922.35072)], they showed that this problem has a self-similar solution, and that \(B\) converges to this self-similar solution as \(t\to\infty\) provided the initial data approach this self-similar solution fast enough as \(x\) and \(|z|\) tend to infinity. In this paper, they study the behavior of solutions with respect to the parameter \(K\). Setting \(\varepsilon=1/K^2\), and using the new variables \(x=X/(t+1)^{1/2}\), \(z= Z/(K(t+1)^{1/2})\), the self-similar problem associated to the original initial-boundary value problem becomes \[ \begin{aligned} -U^\varepsilon_{xx}- \varepsilon U^\varepsilon_{zz}-\frac 12 (zU^\varepsilon_z +xU^\varepsilon_x) &=0 \quad\text{ for } x>0,\;z\in \mathbb R, \\ U^\varepsilon_x=U^\varepsilon U^\varepsilon_z &=0 \quad\text{ for } x=0, z\in \mathbb R, \\ U^\varepsilon(-\infty,x)=1,\;U^\varepsilon(\infty,x) &=0 \quad\text{ for } x>0. \end{aligned} \] The authors show that this problem has a unique solution, and that, as \(\varepsilon\to 0\), \(U^\varepsilon\) converges in \(L^1_{loc}\) to the solution \(U\) of the corresponding problem with \(\varepsilon=0\). Moreover, \(U(x,z)=1\) for \(z<0\) and \(x>0\), and there is a number \(z_0\) such that \(U(x,z)=0\) for \(z>z_0\) and \(x>0\). In addition, various continuity properties are shown for \(U\), including discontinuities along the axis \(z=0\), \(x>0\).

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B25 Singular perturbations in context of PDEs
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