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Solutions to Second-Order Nonlocal Evolution Equations Governed by Non-Autonomous Forms

Sajid Ullah, Vittorio Colao

TL;DR

The paper develops existence and $L^2$-maximal regularity results for second-order non-autonomous evolution equations with nonlocal initial data in a Lions-type sesquilinear-form framework. It leverages fundamental solutions, evolution families, and fixed-point arguments to treat both undamped and damped cases, establishing robust regularity for solutions. The theory is then specialized to model PDEs, including non-autonomous vibrating membranes and memory-dependent diffusion, with detailed verification of hypotheses in the Neumann and memory-context applications. The work provides a systematic method to obtain well-posedness and maximal regularity, offering a bridge between abstract form theory and concrete PDEs with memory effects.

Abstract

Our main contributions include proving sufficient conditions for the existence of solution to a second order problem with nonzero nonlocal initial conditions, and providing a comprehensive analysis using fundamental solutions and fixed-point techniques. The theoretical results are illustrated through applications to partial differential equations, including vibrating viscoelastic membranes with time-dependent material properties and nonlocal memory effects.

Solutions to Second-Order Nonlocal Evolution Equations Governed by Non-Autonomous Forms

TL;DR

The paper develops existence and -maximal regularity results for second-order non-autonomous evolution equations with nonlocal initial data in a Lions-type sesquilinear-form framework. It leverages fundamental solutions, evolution families, and fixed-point arguments to treat both undamped and damped cases, establishing robust regularity for solutions. The theory is then specialized to model PDEs, including non-autonomous vibrating membranes and memory-dependent diffusion, with detailed verification of hypotheses in the Neumann and memory-context applications. The work provides a systematic method to obtain well-posedness and maximal regularity, offering a bridge between abstract form theory and concrete PDEs with memory effects.

Abstract

Our main contributions include proving sufficient conditions for the existence of solution to a second order problem with nonzero nonlocal initial conditions, and providing a comprehensive analysis using fundamental solutions and fixed-point techniques. The theoretical results are illustrated through applications to partial differential equations, including vibrating viscoelastic membranes with time-dependent material properties and nonlocal memory effects.

Paper Structure

This paper contains 8 sections, 16 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abstract Wave Equation
  4. Damped Wave Equation
  5. Applications
  6. Undamped Wave Equation with Neumann Boundary Conditions and Nonlocal Initial Data
  7. Population dynamics with memory effects
  8. Conclusions

Key Result

Theorem 2.1

Let $a$ satisfy the conditions $(A_1)$-$(A_4)$ and let $A(t)$ be the realization of $a(t, \cdot, \cdot)$ in $H$. Assume that the square root property $(S)$ holds. Then for every $f\in L^2(0,T;H)$ and $u_0\in V$, there exists a unique solution $u\in MR(V, H)$ to the problem eq:firstorder.

Theorems & Definitions (30)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3: Pazzy, Chapter 5
  • Definition 2.4: Budde, Kozak
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 20 more