[Paper Review] Existence of solutions for a fractional semilinear parabolic equation with singular initial data

Kotaro Hisa, Kazuhiro Ishige|arXiv (Cornell University)|Jul 5, 2016

Nonlinear Partial Differential Equations14 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for the existence of solutions to a fractional semilinear parabolic equation with singular initial data, using capacity-based estimates and heat kernel analysis. It provides optimal lifespan estimates for solutions with initial data decaying like $|x|^{-A}$ as $\lambda \to 0^+$, refining classical Fujita-type results for fractional Laplacians and extending solvability theory to singular measures and functions with critical decay rates.

ABSTRACT

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem $$ \partial_t u+(-Δ)^{\fracθ{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad u(0)=μ\ge 0\quad\mbox{in}\quad{\bf R}^N, $$ where $N\ge 1$, $01$ and $μ$ is a Radon measure or a measurable function in ${\bf R}^N$. Our conditions lead optimal estimates of the life span of the solution with $μ$ behaving like $λ|x|^{-A}$ ($A>0$) at the space infinity, as $λ o +0$.

Motivation & Objective

• To determine necessary and sufficient conditions on initial data for the solvability of the Cauchy problem involving the fractional Laplacian $(-\Delta)^{\theta/2}$ with $p>1$.
• To extend the classical Fujita-type theory to fractional diffusion by analyzing initial data with singular behavior at infinity.
• To provide optimal estimates for the lifespan of solutions when the initial data $\mu = \lambda \phi$ with $\phi(x) \sim |x|^{-A}$ and $\lambda \to 0^+$.
• To establish the existence and uniqueness of the initial trace for nonnegative solutions, generalizing results from the local case ($\theta = 2$) to fractional orders.

Proposed method

• Use of the fundamental solution $G(x,t)$ of the linear fractional parabolic equation to define weak solutions via integral formulation.
• Application of capacity estimates from potential theory (Meyers' capacity) to derive necessary conditions on initial data for solvability.
• Derivation of sufficient conditions via $L^{r,\infty}$-norm estimates of initial data in balls, adapted to the fractional diffusion scaling.
• Analysis of the behavior of solutions under scaling, particularly for initial data $\mu(x) \sim \lambda |x|^{-A}$, using homogeneity and logarithmic corrections.
• Use of comparison principles and energy-type estimates to bound the solution lifespan from above and below.
• Establishment of sharp lifespan estimates by balancing the growth of the nonlinear term $u^p$ against the decay of the initial data and the fractional diffusion rate.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on a Radon measure or measurable function $\mu$ for the existence of a nonnegative solution to the fractional semilinear parabolic equation $\partial_t u + (-\Delta)^{\theta/2}u = u^p$?
• RQ2 How does the lifespan $T(\lambda\phi)$ of solutions behave as $\lambda \to 0^+$ when the initial data $\phi(x) \sim |x|^{-A}$ at infinity?
• RQ3 Can the classical Fujita-type critical exponent $p_* = 1 + 2/N$ be generalized to the fractional case $0 < \theta \leq 2$, and what role does $\theta$ play in the critical decay rate?
• RQ4 What is the precise dependence of the solution lifespan on the decay rate $A$ of the initial data, especially in the borderline cases $p = p_\theta$ and $A = \theta/(p-1)$?
• RQ5 How do logarithmic corrections in the initial data affect the blow-up time when $A = N$ or $A = \theta/(p-1)$?

Key findings

• For $p > p_\theta = 1 + \theta/N$, the Cauchy problem admits a local-in-time solution if the initial data $\mu$ satisfies $\sup_{x \in \mathbb{R}^N} \int_{B(x,\sigma)} \mu(y)\,dy \leq \gamma \sigma^{N - \frac{2}{p-1}}$ for small $\sigma > 0$, with $\gamma$ depending on $N$, $p$, and $\theta$.
• When $p = p_\theta$, a necessary condition for solvability is $\sup_{x \in \mathbb{R}^N} \mu(B(x,\sigma)) \leq \gamma |\log \sigma|^{-N/\theta}$ for small $\sigma > 0$, generalizing the classical logarithmic capacity condition.
• For initial data $\mu(x) = \lambda \phi(x)$ with $\phi(x) \sim |x|^{-A}$, the lifespan $T(\lambda\phi)$ satisfies $T(\lambda\phi) \geq C \lambda^{-\left(\frac{1}{p-1} - \frac{1}{\theta} \min\{A,N\}\right)^{-1}}$ when $A \neq N$, and a logarithmic correction appears when $A = N$, showing sharp dependence on decay rate.
• In the critical case $p = p_\theta$ and $A = N$, the lifespan satisfies $\log T(\lambda\phi) \leq C \lambda^{-\frac{p-1}{p}}$, indicating a double-logarithmic blow-up time scaling.
• When $1 < p < p_\theta$, the lifespan is bounded above by $T(\lambda\phi) \leq C_2 \left( \frac{\lambda^{-1}}{\log \lambda^{-1}} \right)^{\left(\frac{1}{p-1} - \frac{N}{\theta}\right)^{-1}}$ if $A = N$, showing a slower blow-up than in the supercritical case.
• The paper proves that the initial trace of any nonnegative solution is a unique Radon measure, and provides a complete characterization of the initial data for which solutions exist, even in the singular case $\mu \in L^1_{\text{loc}} \setminus L^\infty$.