arXiv subjects

Mizuki Kojima

Publications and source records attributed to Mizuki Kojima.

4 recordsLinked to original sources

arXiv2024

On a Fujita critical time-fractional semilinear heat equation in the uniformly local weak Zygmund type space

In this paper, we derive sufficient conditions on initial data for the local-in-time solvability of a time-fractional semilinear heat equation with the Fujita exponent in a uniformly local weak Zygmund type space. It is known that the time-fractional problem with the Fujita exponent in the scale critical space $L^1(\mathbb{R}^N)$ exhibits the local-in-time solvability in contrast to the unsolvability of the Fujita critical classical semilinear heat equation. Our new sufficient conditions take into account the fine structure of singularities of the initial data, in order to show a natural correspondance between the time-fractional and the classical case for the local-in-time solvability. We also apply our arguments to life span estimates for some typical initial data.

math.AP↗

arXiv2024

On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

We are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$. \[ {\rm (P)}_α\qquad \partial_t^αu -Δu = u^p,\quad t>0,\,\,\, x\in{\bf R}^N, \qquad u(0) = μ\quad \mbox{in}\quad {\bf R}^N, \] where $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$, $p>1$, and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. The case $α=1$ formally gives the Fujita-type equation (P)$_1$ \ $\partial_tu-Δu=u^p$. In particular, we mainly focus on the Fujita critical case where $p=p_F:=1+2/N$. It is well known that the Fujita exponent $p_F$ separates the ranges of $p$ for the global-in-time solvability of (P)$_1$. In particular, (P)$_1$ with $p=p_F$ possesses no global-in-time solutions, and does not locally-in-time solvable in its scale critical space $L^1(\mathbf{R}^N)$. It is also known that the exponent $p_F$ plays the same role for the global-in-time solvability for (P)$_α$. However, the problem (P)$_α$ with $p=p_F$ is globally-in-time solvable, and exhibites local-in-time solvability in its scale critical space $L^1(\mathbf{R}^N)$. The purpose of this paper is to clarify the collapse of the global and local-in-time solvability of (P)$_α$ as $α$ approaches $1-0$.

math.AP↗

arXiv2023

On solvability of a time-fractional doubly critical semilinear equation, and its quantitative approach to the non-existence result on the classical counterpart

We study a time-fractional semilinear heat equation $$\partial^α_t u -Δu = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$. Since the space $L^1(\mathbb{R}^N)$ is scale critical with $p=1+2/N$, this type of equation is known as a doubly critical problem. It is known that the usual doubly critical equation $\partial_t u-Δu=u^p$ does not have nonnegative global-in-time solutions, while the time-fractional problem does. Moreover, there exists a singular initial data which admits no local-in-time solution, while the time-fractional equation is solvable for any $L^{1}(\mathbb{R}^N)$ initial data. In this paper, we deduce a necessary condition imposed on $u_0$ for the existence of a nonnegative solution. Furthermore, we obtain corollaries that describe the collapse of the local and global solvability for the time-fractional equation as $α\rightarrow 1$.

math.AP↗

arXiv2021

The existence and regularity theory for abstract semilinear time-fractional evolution equations

In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces enables us to deduce space regularity result easily.

math.AP↗