Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients
The author discusses solutions to an initial-boundary-value problem for the one-dimensional diffusion equation \[ \partial_ tu(t,x) = (1/2)\partial_ x^ 2u(t,x) + k(x)\partial_ xu(t,x) + F(u(t,x))\text{ on } {\mathbb{R}}^+\times {\mathbb{R}}^+. \] Investigation of this problem can be motivated by looking at spherically symmetric solutions to a corresponding multidimensional problem: \[ \partial_ tU(t,x) = (1/2)\Delta_ xU(t,x) + F(U(t,x)),\quad (t,x)\in {\mathbb{R}}^+\times {\mathbb{R}}^ N. \] Under various conditions on F the author proves that solutions which blow up behave asymptotically like ''travelling fronts'' i.e. \(u(t,x)\sim w(x+m(t))\) \((t\to \infty)\) where w is a ''travelling wave'' i.e. a special solution to \[ (1/2)w''(x)+cw'(x)+F(w(x))=0. \]
scientific article; zbMATH DE number 4002642