Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Blow-up of solutions for discrete semilinear wave equation with the scale-invariant damping

Koji Wada, Kyouhei Wakasa

TL;DR

This paper analyzes blow-up phenomena for the discretized semilinear wave equation with scale-invariant damping in the subcritical regime 1<p≤1+2/d. Building on Matsuya's discrete blow-up framework, it introduces a discretization that mirrors the continuous equation’s dynamics and proves finite-time blow-up under mild initial-data conditions, demonstrating sharpness in low dimensions d=1,2. The approach combines lattice-point estimates, iterative lower bounds on the total amplitude, and a nonlinear recurrence framework to show that any global-in-time solution is impossible in the discrete setting. By aligning the discrete blow-up threshold with the continuous critical exponents p_F(d) and p_S(d+2), the work bridges discrete and continuous behavior in the transient regime of scale-invariant damping.

Abstract

We consider the blow-up problem for discretized scale-invariant nonlinear dissipative wave equations. It is known that the critical exponents for undiscretized equations (continuous equations) are given by Fujita and Strauss exponents depending on the space dimensions. Our purpose is to obtain results for the discretized equations that correspond to those shown for the continuous one. The proof is based on Matsuya [6], who showed the blow-up problem for discrete semilinear wave equations without dissipative terms, and we found that the result is sharp in the case of one and two space dimensions compared to the continuous equations.

Blow-up of solutions for discrete semilinear wave equation with the scale-invariant damping

TL;DR

This paper analyzes blow-up phenomena for the discretized semilinear wave equation with scale-invariant damping in the subcritical regime 1<p≤1+2/d. Building on Matsuya's discrete blow-up framework, it introduces a discretization that mirrors the continuous equation’s dynamics and proves finite-time blow-up under mild initial-data conditions, demonstrating sharpness in low dimensions d=1,2. The approach combines lattice-point estimates, iterative lower bounds on the total amplitude, and a nonlinear recurrence framework to show that any global-in-time solution is impossible in the discrete setting. By aligning the discrete blow-up threshold with the continuous critical exponents p_F(d) and p_S(d+2), the work bridges discrete and continuous behavior in the transient regime of scale-invariant damping.

Abstract

We consider the blow-up problem for discretized scale-invariant nonlinear dissipative wave equations. It is known that the critical exponents for undiscretized equations (continuous equations) are given by Fujita and Strauss exponents depending on the space dimensions. Our purpose is to obtain results for the discretized equations that correspond to those shown for the continuous one. The proof is based on Matsuya [6], who showed the blow-up problem for discrete semilinear wave equations without dissipative terms, and we found that the result is sharp in the case of one and two space dimensions compared to the continuous equations.

Paper Structure

This paper contains 4 sections, 3 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Discretization of the equations and Main Theorem
  3. Preliminaries
  4. Proof of Theorem 1

Key Result

Theorem 1

Let $d\ge1$, and let $1 < p \leq 1 + 2/d$. Assume that hold for any $R \in {\bf N}$, where $i=(i_1,i_2,\ldots,i_d) \in {\bf Z}^d$ and $||i||_1 := |i_1| + |i_2| + \cdots + |i_d|$. Moreover, we assume that there exists a positive constant $I$ such that Then the solution for (eq:4) blows up in finite time.

Theorems & Definitions (7)

  • Theorem 1
  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof