Abstract
We consider the nonlinear diffusion equationu t −a(x, u x x )+b(x, u)=λg(x, u) with initial boundary conditions\(u\left( {0, x} \right) = \mathop u\limits_ - \left( x \right)\) andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only ifλ∈(0,λ *) orλ∈(0,λ *]. The endpointλ * can be estimated by\(\begin{gathered} \lambda * \leqq \sup \mu _1 \left\{ u \right\} \hfill \\ u \hfill \\ \end{gathered} \), whereμ 1 u denotes the first eigenvalue of the stationary problem linearized at the “point”u. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.
Zusammenfassung
Wir betrachten die nichtlineare Diffusionsgleichungu t −a(x, u x ) x +b(x, u)=λg(x, u) mit Randbedingungen\(u\left( {0, x} \right) = \mathop u\limits_ - \left( x \right)\) undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, fallsλ∈ (0,λ *) oderλ∈(0,λ *]. Der Endpunktλ * kann durch\(\begin{gathered} \lambda * \leqq \sup \mu _1 \left\{ u \right\} \hfill \\ u \hfill \\ \end{gathered} \) abgeschätzt werden, wobeiμ 1 u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.
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Weyer, J. On the positive solutions of some nonlinear diffusion problems. Z. angew. Math. Phys. 36, 499–507 (1985). https://doi.org/10.1007/BF00945292
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DOI: https://doi.org/10.1007/BF00945292