Hiroshi Matano

Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations

Abstract We consider the initial value problem for the semilinear heat equation u t = u xx + f ( u , t ) (0 x L , t > 0) under the Dirichlet, the Neumann, or the periodic boundary conditions. We show that each solution—whether it exists globally for t > 0 or blows up in a finite time—possesses an “asymptotic profile” in a certain sense and tends to this profile as time increases. In the special case where f ( u , t + T ) ≡ f ( u , t ) for some T > 0, among other things, the above statement is interpreted as saying that any bounded global solution converges as t → ∞ to a time T -periodic solution having some specific spatial structure. In the case where the solution blows up in a finite time (say at t = t 0 ), assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function, we prove that the blow-up set is a finite set and that lim t ↑ t 0 u ( x , t ) = ϑ ( x ) exists, with ϑ being a smooth function having at most finitely many singular points.

Large time behavior of solutions of a dissipative semilnear heat equation

In this paper we investigate the large time behavior of solutions of the semilinear heat equation. Where u{sub 0} is the initial data, N {ge} 1 and p > 1. It can be easily checked that if u(t,x) satisfied (1.1), then for {gamma} > 0 the rescaled functions u{sub {gamma}}(t,x) satisfies (1.1), then for {gamma}>0 the rescaled functions define a one parameter family of solutions to (1.1). A solution u {equivalent_to} 0 is said to be self-similar, when u{sub {gamma}} {equivalent_to} u for all {gamma} > 0. For instance, for any fixed p > 1, w{sup *}(t,x):=((p-1)t){sup {minus}1/(p-1)} is such a solution. Actually, it has been proved by H.Brezis, L.A. Peletier & D. Terman that for 1 {infinity}. 15 refs.

Convergence and sharp thresholds for propagation in nonlinear diffusion problems

We study the Cauchy problem utDuxxCf.u/ .t > 0; x2 R 1 /; u.0;x/Du0.x/ .x2 R 1 /; wheref.u/ is a locally Lipschitz continuous function satisfyingf.0/D 0. We show that any non- negative bounded solution with compactly supported initial data converges to a stationary solution as t! 1. Moreover, the limit is either a constant or a symmetrically decreasing stationary solu- tion. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solutionu , we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even iff has a jumping discontinuity atuD 1.

Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations

We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities – including multistable ones – and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ℝ n , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ℝ n . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.

Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit

We study a curvature-dependent motion of plane curves in a two-dimensional cylinder with periodically undulating boundary. The law of motion is given by

IMMEDIATE REGULARIZATION AFTER BLOW-UP

V=\kappa + A

Singular Limit of a Reaction-Diffusion Equation with a Spatially Inhomogeneous Reaction Term

, where

Chapter 14 – Blow-up in Nonlinear Heat Equations from the Dynamical Systems Point of View

V

Stability analysis in order-preserving systems in the presence of symmetry

is the normal velocity of the curve,