Seminar 27.10.2022: Jin Takahashi a Tatsuki Kawakami

[quittner]

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Štvrtok 27.10.2022 o 14:00, poslucháreň M-223

Jin Takahashi (Tokyo Institute of Technology):
Blow-up of the critical norm for a supercritical semilinear heat equation

Abstract:
We consider the scaling critical Lebesgue norm of blow-up solutions
to a semilinear heat equation (the Fujita equation) in an arbitrary
smooth domain. In the Sobolev supercritical range, we show that
the critical norm must be unbounded near the blow-up time.
The range is optimal in view of the existence of blow-up solutions
with bounded critical norm for the Sobolev critical case.

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Tatsuki Kawakami (Ryukoku University):
Existence of solutions to nonlinear parabolic equations
via majorant integral kernel

Abstract:
In this talk, we consider the nonlinear integral equation
with an integral kernel, which is a generalization
of the fundamental solutions to the heat equation,
fractional diffusion equations, and higher-order parabolic equations.
By using a majorant kernel, which is based on the fundamental solution
to the fractional diffusion equation, we obtain sufficient conditions
for the existence of solutions to the integral equation
under a suitable structure condition on the nonlinear term.
Our main results are applicable to the Cauchy problem for various nonlinear
parabolic equations including fractional semilinear parabolic equations,
higher-order semilinear parabolic equations,
and viscous Hamilton-Jacobi equations.

This talk is based on several joint works with K. Ishige and S. Okabe.