Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders
This paper is concerned with the existence of traveling wave solutions of reaction-diffusion systems of the gradient type \(u_t=\Delta u-\nabla_uV(u)\), where \(u=u(x,t)\). After introducing the functional framework, the authors define the main variational problem associated to this nonlinear parabolic system. Next, it is established the existence of solutions, viewed as minimizers of the associated energy functional. A particular attention is given to the boundedness and regularity of solutions, as well as to global gradient estimates for the minimizers. Uniform estimates allow the authors to obtain information on the exponential decay of functions obeying related estimates of the norm of solutions. In the last part of the present paper there are deduced several criteria of existence and non-existence of traveling waves. There are also established several properties of minimizers, such as their one-dimensionality in the case of Neumann boundary conditions, or the fact that for potentials \(V\) depending only on the magnitude of the vector \(u\), the minimizers are scalar (up to a constant vector). Several applications of the abstract results by using a computer-assisted approach conclude the present paper.
scientific article; zbMATH DE number 5294564
scientific article