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Singular reaction-diffusion boundary value problems. (English) Zbl 0815.35019
The paper deals with nonnegative solutions of the one-dimensional boundary value problem: \[ y'' + f(x,y') = \Phi^ 2 g(x,y) \quad \text{for} \quad a < x < b, \qquad (2) \quad y'(a) = 0,\;\beta y'(b) + \alpha y(b) = A \tag{1} \] with \(\beta \geq 0\) and \(\alpha, A > 0\). Problems of this kind arise by studying the radial solutions of the reaction- diffusion equation \(\Delta u = \Phi^ 2v (x,u)\) with \(x \in \Omega \subset \mathbb{R}^ N\) and \(v\) having radial symmetry with respect to \(x\). Here \(u \geq 0\) is the concentration of one of the reactants and \(\Phi^ 2\) denotes the so-called Thiele modulus. In (1) \(g\) is allowed to be unbounded for \(y \to 0+\). The authors prove several theorems concerning the existence of the strictly positive solutions to (1), (2) or the so- called dead core solutions (vanishing on an interval \([a,x_ 0]\), \(a \leq x_ 0 < b)\). They also consider the case when \(g\) may be approximated by some more regular \(g_ k\). The paper is closely related to the earlier papers of L. E. Bobisud [J. Math. Anal. Appl. 147, No. 1, 249-262 (1990; Zbl 0706.34052); and J. Differ. Equations 85, No. 1, 91-104 (1990; Zbl 0704.34033)] and extends some results of them.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35K57 Reaction-diffusion equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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