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Local well-posedness for a class of semilinear Moore-Gibson-Thompson equations with subcritical nonlinearities

Flank D. M. Bezerra, Luís M. Salge

TL;DR

This work addresses local well-posedness for a class of Moore-Gibson-Thompson type equations with a $2m$-order elliptic operator and subcritical nonlinearities. By recasting the problem as a semilinear evolution on a product space and employing sectorial and extrapolation-space theory, the authors overcome the lack of accretivity and bounded imaginary powers of the linear part. They prove a local existence and uniqueness result in extrapolated fractional spaces $Y_{(-1)}^{\alpha}$ for $\alpha$ near $1$, with enhanced regularity for smoother initial data. The analysis clarifies the role of extrapolation and fractional domains in high-order PDE well-posedness on bounded domains and provides a framework applicable to MGT-type models and similar higher-order semilinear problems.

Abstract

In this paper, we study a class of higher-order semilinear evolution equations inspired by the Moore-Gibson-Thompson model introduced by Dell'Oro, Liverani and Pata (2023), involving strongly elliptic operators of order ($2m$) with homogeneous boundary conditions. The associated unbounded linear operator is a sectorial operator with zero belonging to the resolvent set, allowing the construction of fractional powers spaces, and the analysis of their spectral properties. We prove the local well-posedness of the corresponding semilinear Cauchy problem under subcritical nonlinearities. Our framework clarifies the role of extrapolation spaces and fractional domains in handling the lack of accretivity and bounded imaginary powers of the operator.

Local well-posedness for a class of semilinear Moore-Gibson-Thompson equations with subcritical nonlinearities

TL;DR

This work addresses local well-posedness for a class of Moore-Gibson-Thompson type equations with a -order elliptic operator and subcritical nonlinearities. By recasting the problem as a semilinear evolution on a product space and employing sectorial and extrapolation-space theory, the authors overcome the lack of accretivity and bounded imaginary powers of the linear part. They prove a local existence and uniqueness result in extrapolated fractional spaces for near , with enhanced regularity for smoother initial data. The analysis clarifies the role of extrapolation and fractional domains in high-order PDE well-posedness on bounded domains and provides a framework applicable to MGT-type models and similar higher-order semilinear problems.

Abstract

In this paper, we study a class of higher-order semilinear evolution equations inspired by the Moore-Gibson-Thompson model introduced by Dell'Oro, Liverani and Pata (2023), involving strongly elliptic operators of order () with homogeneous boundary conditions. The associated unbounded linear operator is a sectorial operator with zero belonging to the resolvent set, allowing the construction of fractional powers spaces, and the analysis of their spectral properties. We prove the local well-posedness of the corresponding semilinear Cauchy problem under subcritical nonlinearities. Our framework clarifies the role of extrapolation spaces and fractional domains in handling the lack of accretivity and bounded imaginary powers of the operator.

Paper Structure

This paper contains 6 sections, 7 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main problem
  3. Definitions and Notations
  4. Spectral properties of the linear operators
  5. Local well posedness with subcritical nonlinearities
  6. Statements and Declarations

Key Result

Theorem 2.2

If the linear operator B is sectorial, and $F : Z^{\eta} \to Z$ is Lipschitz continuous in bounded subsets of $Z^\eta$ with $\eta\in[0,1)$, then, given $r > 0$ there exists a $T_0 > 0$ and for each $z_0 \in Z^{\eta}$ with $\|z_0\|_{Z^{\eta}} \leq r$ a function $z(\cdot, z_0) \in C([0, T_0], Z^\eta) is continuous; $z(\cdot, z_0)$ is the unique solution of abstrcauchyp.

Figures (1)

  • Figure 1: Partial description of the fractional power spaces scales

Theorems & Definitions (18)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • ...and 8 more