Semilinear Equations Including the Mixed Operator

TL;DR

The paper analyzes the Cauchy problem for a semilinear evolution equation with a mixed local-nonlocal operator $\mathcal{L} = -\Delta + (-\Delta)^{\alpha/2}$, $0<\alpha<2$, driven by a nonlinear term $-h(t)u^p$ and gradient-free initial data. Using a mild-solution framework built on the heat kernel $E_\alpha$ and a contraction-mapping argument, it proves local existence and uniqueness in $L^{\infty}$ and establishes a blow-up alternative via a maximal existence time $T_m$, along with nonnegativity and a global existence criterion under suitable data. The work further develops regularity theory: Schauder-type results yield higher regularity of solutions and, under appropriate initial-data smoothness, classical global solutions or $W^{2,1,r}$-type regularity, linking the nonlinear dynamics to the diffusion generated by the mixed operator. These results advance understanding of semilinear equations with mixed local-nonlocal diffusion and provide tools for analyzing global behavior and regularity of solutions.

Abstract

We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -Δ+ (-Δ)^{α/2} \), where \( 0 < α< 2 \). The Cauchy problem under consideration is \begin{equation*} \partial_t u + t^βL u = -h(t) u^p, \quad x \in \mathbb{R}^N, \quad t > 0, \end{equation*} with nonnegative initial data \( u(x, 0) = u_0(x) \). We establish the existence and uniqueness of local solutions in \( L^\infty(\mathbb{R}^N) \) using a contraction mapping argument. Furthermore, we analyze conditions for global existence, proving that solutions remain globally bounded in time under appropriate assumptions on the parameters \( β\), \( p \), and the function \( h(t) \).

Theorems & Definitions (23)

• Definition 1
• Proposition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Definition 2: Mild solution