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Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity

Dhouha Draouil, Mohamed Majdoub

Abstract

We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy space for radially symmetric initial data. Additionally, we derive the linearization of energy-bounded solutions. The main challenge in our analysis arises from the spatial growth of the nonlinearity at infinity, which prevents the direct application of Sobolev embeddings or Hardy inequalities to control the potential energy. The main novelty in this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.

Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity

Abstract

We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy space for radially symmetric initial data. Additionally, we derive the linearization of energy-bounded solutions. The main challenge in our analysis arises from the spatial growth of the nonlinearity at infinity, which prevents the direct application of Sobolev embeddings or Hardy inequalities to control the potential energy. The main novelty in this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.
Paper Structure (11 sections, 14 theorems, 91 equations)

This paper contains 11 sections, 14 theorems, 91 equations.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Useful tools & Auxiliary results
  4. Global well posedness
  5. Proof of Theorem \ref{['main2D']}
  6. Local Existence
  7. Global existence
  8. Proof of Theorem \ref{['main3D']}
  9. Linearization
  10. The 2D case
  11. The 3D case

Key Result

Theorem 1.1

Let $N=2$, $0\leq b\leq 1/2$, $\bm=1$, and suppose that the nonlinearity $f$ is given by 2D. Then, for any radially symmetric data $(u_0,u_1)\in H^{1}\times L^{2}$, the initial value problem eq a1-eq aC.D has a unique global solution $u$ in the space ${\mathcal{C}}(\mathbb{R},H^{1})\cap {\mathcal{C}

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['L-1']}
  • Lemma 2.4
  • ...and 15 more