Related papers: Convergence of Rothe's method for fully nonlinear …

In this paper, we investigate the inverse problem of determining an unknown time-dependent source term in a semilinear pseudo-parabolic equation with variable coefficients and a Dirichlet boundary condition. The unknown source term is…

We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a…

We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ in $M\times(0,\infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the…

In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and…

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial…

For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…

In this article we develop an analogue of Aubry Mather theory for time periodic dissipative equation \[ \left\{ \begin{aligned} \dot x&=\partial_p H(x,p,t),\\ \dot p&=-\partial_x H(x,p,t)-f(t)p \end{aligned} \right. \] with $(x,p,t)\in…

The main objective of this paper and the accompanying one \cite{ETZ2} is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work \cite{EKTZ}, focused on the…

In this work, we develop weighted Lorentz-Sobolev estimates for viscosity solutions of fully nonlinear elliptic equations with oblique boundary condition under weakened convexity conditions in the following configuration $F(D^{2}u, Du, u,…

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is $C^{2,\alpha}$ on the compliment of a closed set of Hausdorff dimension at most $\epsilon$ less than the dimension. The equation is assumed to be $C^1$,…

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical…

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and…

We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-\Delta_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity…

We consider positive solutions of cooperative parabolic Lotka-Volterra systems with equal diffusion coefficients, in bounded and unbounded domains. The systems are complemented by the Dirichlet or Neumann boundary conditions. Under suitable…

We show the existence of continuous viscosity solutions to the equation describing the flow of a graph in the Carnot group G x R according to its horizontal Gauss curvature. In doing so, we prove a comparison principle for degenerate…

We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases:…

In the paper we prove the convergence of viscosity solutions $u_{\lambda}$ as $\lambda\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi equation \[ H(x,d_x u, \lambda u)=\alpha(x)\Delta u,\quad \alpha(x)\geq 0,\quad…

We address the persistence of regularity for the 2D $\alpha$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_{0}, \rho_{0}) \in W^{s,q}(\mathbb R^2) \times…

In this paper, we consider the following Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,t,u(x,t),\partial_xu(x,t))=0, u(x,0)=\phi(x). \end{cases} \end{equation*} Under some assumptions…

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation $$ \partial_t u^q-\mbox{div}(|Du|^{p-2}Du)=0 $$ in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p)_+}$. We show that when solutions…