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Multidimensional reaction diffusion equations with nonlinear boundary conditions. (English) Zbl 0932.35120
Extending the work in A. A. Lacey, J. R. Ockendon, J. Sabina and D. Salazar [Rocky Mt. J. Math. 26, No. 1, 195-212 (1996; Zbl 0858.35066)] this paper is concerned with the asymptotic behaviour or \(\varepsilon\to 0^+\) of solutions of the weakly nonlinear boundary value problem \[ {\partial u\over\partial t}=\Delta u-\lambda uf(\varepsilon \ln u),\quad u(x,0)= u_0(x) \] in a bounded domain \(\Omega\) with piecewise smooth boundary \(\partial\Omega\), on the smooth parts of which \[ {\partial u\over\partial n}= ug(\varepsilon \ln u), \] where \(\partial/\partial n\) denotes the outward normal derivative. Here \(\lambda\) is a positive constant, and the dependence and stability of the solutions on \(\lambda\) is investigated. This dependence is revealed (in the limit of vanishing \(\varepsilon\)) via the solution \(v\) of the ordinary differential equation \[ {dv\over dt}+\lambda f(v)= F(g(v)) \] related to \(u\) by \(u\sim w(x,\varepsilon t)\exp(v(\varepsilon t)/\varepsilon)\). Here \(F(\gamma)\) denotes the largest eigenvalue of \[ \Delta\phi= \Lambda\phi\quad\text{in }\Omega,\quad {\partial\phi\over\partial n}= \gamma\phi\quad\text{on }\partial\Omega. \] The authors’ investigation of this eigenvalue problem is principally by variational means. Their results are applied to various applications of exponential \(f\) and \(g\).

35K57 Reaction-diffusion equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Zbl 0858.35066
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