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We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ partial_t u + (-Delta)^{ 1/2} u = | abla u|^p, quad x in mathbb R^N, t > 0, qquad u(x,0) = u_0(x) , quad x in mathbb R^N, $$ where $p > 1$ and $u_0 in B^1_{r,1}(mathbb R^N) cap B^1_{infty,1} (mathbb R^N)$ with $r in [1,infty]$. We show that for sufficiently small $u_0 in dot B^1_{infty,1}(mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.
In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on
We prove existence and uniqueness of Crandall-Lions viscosity solutions of Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated to the optimal control of path-dependent SDEs. This seems the first uniqueness result in such a
Let $(Omega, mu)$ be a probability space endowed with an ergodic action, $tau$ of $( {mathbb R} ^n, +)$. Let $H(x,p; omega)=H_omega(x,p)$ be a smooth Hamiltonian on $T^* {mathbb R} ^n$ parametrized by $omegain Omega$ and such that $ H(a+x,p;tau_aomeg
The paper is devoted to prove the existence of a local solution of the Hamilton-Jacobi equation in field theory, whence the general solution of the field equations can be obtained. The solution is adapted to the choice of the submanifold where the in