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Global existence and nonexistence of semilinear wave equation with a new condition

Bolys Sabitbek

Abstract

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition $$α\int_0^{u } f(s)ds \leq uf(u) + βu^2 +ασ,$$ for some positive constants $α$, $β$, and $σ$, where $β< \frac{λ_1(α-2)}{2}$ with $λ_1$ being a first eigenvalue of Laplacian. By introducing a family of potential wells, we establish the invariant sets, vacuum isolation of solutions, global existence and blow-up solutions of semilinear wave equation for initial conditions $E(0)<d$ and $E(0)=d$.

Global existence and nonexistence of semilinear wave equation with a new condition

Abstract

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition for some positive constants , , and , where with being a first eigenvalue of Laplacian. By introducing a family of potential wells, we establish the invariant sets, vacuum isolation of solutions, global existence and blow-up solutions of semilinear wave equation for initial conditions and .
Paper Structure (15 sections, 23 theorems, 134 equations)

This paper contains 15 sections, 23 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Setting problem
  3. A new condition on nonlinearity
  4. Literature overview
  5. Main goal of the paper
  6. Plan of the paper
  7. Potential Wells Theory
  8. Invariant sets and vacuum isolating of solutions
  9. Global existence of solutions for semilinear wave equation
  10. Initial condition $0<E(0)<d$ and $I(u_0)>0$
  11. Critical initial condition $E(0)=d$ and $I(u_0)\geq 0$
  12. Blow-up solutions of semilinear wave equation
  13. Initial condition $E(0)<d$ and $I(u_0)<0$
  14. Critical initial condition $E(0)=d$ and $I(u_0)<0$
  15. Conclusion

Key Result

Lemma 1.1

Let $f(u)$ satisfy $(H)$. There are two cases: $(H)-(a)$$f$ is monotone and is convex for $u>0$, concave for $u<0$, then $(H)-(b)$$f$ is convex for $-\infty < u < + \infty$, then

Theorems & Definitions (42)

  • Lemma 1.1
  • Lemma 1.2
  • proof : Proof of Lemma \ref{['lem']}
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem-1']}
  • Remark 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['lem-2']}
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lem-3']}
  • ...and 32 more