Qualitative theory and asymptotic analysis of nonlinear partial differential equations

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We have considered the behavior of solutions after the blow-up time for nonlinear heat equations with a power nonlinearity and those with an exponential nonlinearity. It is shown that singularities that apper at the blow-up time disappear and the solutions become smooth immediately (Matano, SIAM J.Math.Anal., in press). We have also studied the blow-up rate for nonlinear heat equations with a power nonlinearity and proved that the blow-up is always type 2 so far as the power is in the intermediate supercritical range (Matano, Comm.Pure Appl.Math., 2004).Yamamoto has studied an inverse problem of determining two unknown convection terms in a two-dimensional elliptic equation, and proved that those terms can be determined by the so-called Dirichlet-Neumann map (Inverse Problems, 2004).Weiss has considered a singular limit problem for parabolic equations that are applicable to the double obstacle problem. Using a new monotonicity formula, he has succeeded in exstimating the Hausdorff dimension of the free boundary (Calc.Var.PDE, 2003).Ei has studied a reaction-diffusion system on a two-dimensional cylinder and analysed the behavior of solutions having a pulse-like profile. He derived an equation governing the motion of slow pulses and proved that traveling pulses are mutually repelling (DCDS, Ser.A, in press).Taniguchi has considered the so-called "singular limit eigenvalue problem method" (SLEP method), which is a powerful tool in analyzing the stability of stationary solution of the singular limit problem. He has generalized this method so that it applies to problems in unbounded domains. Using this result, he has proved the stability of planar traveling waves in a bistable reaction-diffusion system (DCDS, Ser.B, 2003).