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Tatsuki Kawakami

Publications and source records attributed to Tatsuki Kawakami.

27 records · Page 2Linked to original sources
arXiv2017

Critical exponent for the global existence of solutions to a nonlinear degenerate/singular parabolic equation

We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.

math.AP↗
arXiv2016

Asymptotic expansions of solutions of fractional diffusion equations

In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ \partial_t u+(-Δ)^{\fracθ{2}}u=0\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), \qquad u(x,0)=φ(x)\quad\mbox{in}\quad{\bf R}^N, $$ where $0<θ<2$ and $φ\in L_K:=L^1({\bf R}^N,\,(1+|x|)^K\,dx)$ with $K\ge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ \partial_t u+(-Δ)^{\fracθ{2}}u=|u|^{p-1}u\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), $$ where $0<θ<2$ and $p>1+θ/N$.

math.AP↗
arXiv2016

Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation

We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted $L^1$ and $L^\infty$ estimates. Furthermore, we establish the higher order asymptotic expansion of the solution. This means that we construct the nonlinear approximation of the global solution with respect to the weight of the data. Our proof is based on the approximation formula of the linear solution, which is given in [36], and the nonlinear approximation theory for a nonlinear parabolic equation developed by [18].

math.AP↗
arXiv2015

Supersolutions for a class of nonlinear parabolic systems

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=Δu+v^p,\qquad & x\inΩ,\,\,\,t>0,\\ \partial_t v=Δv+u^q, & x\inΩ,\,\,\,t>0,\\ u=v=0, & x\in\partialΩ,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\inΩ, \end{array} \right. $$ where $p\ge 0$, $q\ge 0$, $Ω$ is a (possibly unbounded) smooth domain in ${\bf R}^N$ 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.

math.AP↗
arXiv2015

Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces

We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ \partial_t u + (-Δ)^{ 1/2} u = |\nabla u|^p, \quad x \in \mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ where $p > 1$ and $u_0 \in B^1_{r,1}(\mathbb R^N) \cap B^1_{\infty,1} (\mathbb R^N)$ with $r \in [1,\infty]$. We show that for sufficiently small $u_0 \in \dot B^1_{\infty,1}(\mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.

math.AP↗
arXiv2013

Asymptotics for a nonlinear integral equation with a generalized heat kernel

This paper is concerned with a nonlinear integral equation $$ (P)\qquad u(x,t)=\int_{{\bf R}^N}G(x-y,t)φ(y)dy+\int_0^t\int_{{\bf R}^N}G(x-y,t-s)f(y,s:u)dyds, \quad $$ where $N\ge 1$, $φ\in L^\infty({\bf R}^N)\cap L^1({\bf R}^N,(1+|x|^K)dx)$ for some $K\ge 0$. Here $G=G(x,t)$ is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of $(P)$ behaving like a multiple of the integral kernel $G$ as $t\to\infty$.

math.AP↗
arXiv2012

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

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \partial_t u=Δu+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=φ(x)\quad in \quad{\bf R}^N, $$ and assume that the solution $u$ behaves like the Gauss kernel as $t\to\infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $φ$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to\infty$. This paper is a generalization of our previous paper, and our arguments are applicable to the large class of nonlinear parabolic equations.

math.AP↗