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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
##### Keywords:
plasma physics; discontinuities; continuity properties
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##### References:
 [1] Adams, R., Sobolev spaces, (1975), Acad. Press · Zbl 0314.46030 [2] Agmon, S.; Douglis, A.; Nirenberg, L.; Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. pure appl. math., Comm. pure appl. math., 17, 35-92, (1964) · Zbl 0123.28706 [3] Barles, G., Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. diff. equations, Vol. 106, No. 1, 90-106, (1993) · Zbl 0786.35051 [4] Chuvatin, A., Thèse de doctorat de l’école polytechnique, (1994) [5] Crandall, M.G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. amer. soc., 27, 1-67, (1992) · Zbl 0755.35015 [6] Evans, L.C.; Gariepy, R., Measure theory and fine properties of functions, () · Zbl 0804.28001 [7] Gordeev, A.V.; Grechikha, A.V.; Kalda, Y.L., Rapid penetration of a magnetic field into a plasma along an electrode, Sov. J. plasma phys., 16, Vol. 1, 55-57, (1990) [8] Leveque, R.J., Numerical methods for conservation laws, () · Zbl 0682.76053 [9] Lieberman, G.; Trudinger, N., Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. A.M.S., Vol. 295, 509-546, (1986) · Zbl 0619.35047 [10] Lions, P.-L.; Souganidis, P.E., Convergence of MUSCL type methods for scalar conservation laws, C.R. acad. sci. Paris, Vol. 311, 259-264, (1990), Série 1 · Zbl 0712.65082 [11] Méhats, F., Thèse de doctorat de l’école polytechnique, (1997) [12] \scF. Méhats and \scJ.-M. Roquejoffre, A nonlinear oblique derivative boundary value problem for the heat equation. Part 1: Basic results, to appear in Ann. IHP, Analyse Non Linéaire.
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