| Abstract: |
| We give a result regarding wellposedness of solutions to the Coagulation-Fragmentation with multiplicative coagulation and constant fragmentation. This is a critical case where the existence of mass preserving solutions depends on the initial data. Our result resolves partially an open question in the field that there is a critical mass so that the equation is wellposed if and only if the initial data start off below it. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation.We give some results regarding wellposedness and long-time behavior of solutions to the Coagulation-Fragmentation with multiplicative coagulation and constant fragmentation. This is a critical case where the existence of mass preserving solutions depends on the initial data. Our result resolves partially an open question posed by Escobedo, Laurencot, Mischler and Perthame that there is a critical mass so that the equation is wellposed if and only if the initial data start off below it. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. |
|