Tatsuki Kawakami

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

AbstractThis paper is concerned with the Cauchy problem for the nonlinear parabolic equation

Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity

{\partial _t}u| = \vartriangle u + F(x,t,u,\nabla u){\text{ in }}{{\text{R}}^N} \times (0,\infty ),{\text{ }}u(x,0) = \varphi (x){\text{ in }}{{\text{R}}^N},

Positive Solutions of a Semilinear Elliptic Equation with Singular Dirichlet Boundary Data

, where

The decay of the solutions for the heat equation with a potential

\begin{gathered} N \geqslant 1, \hfill \\ F \in C(R^N \times (0,\infty ) \times R \times R^N ), \hfill \\ \phi \in L^\infty (R^N ) \cap L^1 (R^N ,(1 + |x|^K )dx)forsomeK \geqslant 0 \hfill \\ \end{gathered}

Refined asymptotic profiles for a semilinear heat equation

. We give a sufficient condition for the solution to behave like a multiple of the Gauss kernel as t → ∞ and obtain the higher order asymptotic expansions of the solution in W1,q(RN) with 1 ≤ q ≤ ∞.

Global solutions of the heat equation with a nonlinear boundary condition

Abstract In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including { ∂ t u = Δ u + v p , x ∈ Ω , t > 0 , ∂ t v = Δ v + u q , x ∈ Ω , t > 0 , u = v = 0 , x ∈ ∂ Ω , t > 0 , ( u ( x , 0 ) , v ( x , 0 ) ) = ( u 0 ( x ) , v 0 ( x ) ) , x ∈ Ω , where p ≥ 0 , q ≥ 0 , Ω is a (possibly unbounded) smooth domain in R N and both u 0 and v 0 are nonnegative and locally integrable functions in Ω. The supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow-up rate of the solutions.