Boundary Value Problems with Bounded Nonlinearity and General Null-Space of the Linear part.
Bounded solutions of nonlinear parabolic equations.
Bounded weak solutions to nonlinear elliptic equations.
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions : either the space derivative blows up in finite time (with itself remaining bounded), or is global and converges in norm to the unique steady state. The main difficulty is to prove boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...
Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity
This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function and the growth term under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that
Boundedness of the solution of the third problem for the Laplace equation
A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.
Bounds for elliptic operators in weighted spaces.
Bounds for KdV and the 1-d cubic NLS equation in rough function spaces
We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time bounds in terms of the size of the initial data for (joint work with D. Tataru, [15, 14]) , and the solutions to KdV satisfy global a priori estimate in (joint work with T. Buckmaster [2]).