| _version_ | 1866901147021737984 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper rigorously investigates the interplay between critical regularity and wave breaking phenomena in a class of nonlinear D'Alembert equations. We consider initial value problems for semilinear and quasilinear hyperbolic equations, focusing on the conditions on initial data that dictate global existence of smooth solutions versus finite-time singularity formation. Utilizing energy methods, the method of characteristics, and Sobolev space theory, we establish precise critical regularity thresholds below which solutions are guaranteed to exist globally and above which wave breaking, characterized by the blow-up of spatial gradients, is inevitable for suitable initial configurations. Our analysis delineates the specific roles of the nonlinearity's structure and the initial data's smoothness and amplitude in determining the global behavior of solutions. The findings contribute to the broader understanding of well-posedness theory and the dynamics of singularities in nonlinear hyperbolic partial differential equations. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17838833 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Critical Regularity and Wave Breaking in Nonlinear D'Alembert Equations Revista, Zen MATH, 10 This paper rigorously investigates the interplay between critical regularity and wave breaking phenomena in a class of nonlinear D'Alembert equations. We consider initial value problems for semilinear and quasilinear hyperbolic equations, focusing on the conditions on initial data that dictate global existence of smooth solutions versus finite-time singularity formation. Utilizing energy methods, the method of characteristics, and Sobolev space theory, we establish precise critical regularity thresholds below which solutions are guaranteed to exist globally and above which wave breaking, characterized by the blow-up of spatial gradients, is inevitable for suitable initial configurations. Our analysis delineates the specific roles of the nonlinearity's structure and the initial data's smoothness and amplitude in determining the global behavior of solutions. The findings contribute to the broader understanding of well-posedness theory and the dynamics of singularities in nonlinear hyperbolic partial differential equations. |
| title | Critical Regularity and Wave Breaking in Nonlinear D'Alembert Equations |
| url | https://doi.org/10.5281/zenodo.17838833 |