Interface motion and blow-up phenomena in nonlinear partial differential equations

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The aim of this research project is to make a theoretical study of various nonlinear problems related to interfacial motions and blow-up phenomena, by developing asymptotic methods based on the theory of infinite dimensional dynamical systems and stochastic methods, and also performing numerical simulations if necessary. We have obtained the following results.(1) We have given an optimal estimate concerning the singular limit of Allen-Cahn type nonlinear diffusion equations, that has not been known previously. We also obtained similar results for FitzHugh-Nagumo systems and Lotka-Volterra competition systems.(2) We have clarified the global dynamics of blow-up solutions in nonlinear diffusion equations. This was made possible by extending the existing theory on global attractors to the case where blow-up occurs.(3) In nonlinear heat equations with power nonlinearity, the blow-up of solutions is classified into type I and type II, the latter being much more difficult to analyze than the … More former. By using a topological method based on the braid group theory, we have succeeded in determining all possible type II blow-up rates.(4) We studied the stationary problem for the Allen-Cahn equation on the 2-dimensional space having lattice periodicity by using variational methods. We have shown that a necessary and sufficient condition for the existence of multi-layered stationary solution is that the set of single layered solutions has a gap somewhere ; in other words, this set should not be a continuum (foliation).(5) We considered periodic traveling waves in a two-dimensional infinite strip whose boundaries are saw-tooth shaped. We determined the homogenization limit of such traveling waves as one lets the boundary undulation finer and finer.(6) Various partial differential equations are derived from microscopic models via hydrodynamic limit. Those equations include the Stefan problem and some stochastic differential equations.(7) Regularity of free boundaries arising in various elliptic equations such as the combustion model has bee established.(8) Stability of V-shaped traveling wave in the equation of curvature-dependent motion has been established. Less