Related papers: Tug-of-War games and parabolic problems with spati…

We investigate the boundedness and large time behavior of solutions of the Cauchy-Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: $$ u_t = (|u-x|^{p-2} u-x)_x+|u_x|^q \qquad \text{in}\quad…

For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{\beta }u=-(-\Delta )^{\frac{\alpha}{2}}(\rho (v)u),& t>0\\ (-\Delta )^{\frac{\alpha}{2}} v+v=u,& t>0 \end{cases} \end{equation*}…

This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive.…

We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized…

The purpose of this paper is to investigate the time behavior of the solution of a weighted $p$-Laplacian evolution equation, given by \begin{align} \label{eveq} \begin{cases} u_{t} = \text{div} \left(\gamma |\nabla u|^{p-2}\nabla u \right)…

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u &…

In this paper we treat the numerical approximation of the two-phase parabolic obstacle-like problem: \[\Delta u -u_t=\lambda^+\cdot\chi_{\{u>0\}}-\lambda^-\cdot\chi_{\{u<0\}},\quad (t,x)\in (0,T)\times\Omega,\] where $T < \infty, \lambda^+…

We address the problem of existence and (non-)uniqueness of solutions $\big(c,u(\cdot),\mu\big)$ to ergodic mean-field games in the whole space $\mathbb{R}^{m}$ with unbounded and merely measurable data, and for non-separable Hamiltonian.…

We consider a parabolic-elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$ \left\{ \begin{array}{l} u_t -\Delta (u \gamma(v)= \mu u(1-u), \\ - \Delta v +v=u, \end{array} \right. $$…

We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in $C^{2,\alpha}(B_R)\cap C(\overline{B_R})$ for the inhomogeneous $\infty$-Bilaplacian…

In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the…

In this paper, we establish the boundary regularity results for viscosity solutions of fully nonlinear degenerate/singular parabolic equations of the form $$u_t - x_n^{\gamma} F(D^2 u,x,t) = f,$$ where $\gamma<1$. These equations are…

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We…

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$, where $\Omega\subset…

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of a unique viscosity solution of these variational inequalities and give a stochastic representation to…

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N}…

We prove local Lipschitz continuity and Harnack's inequality for value functions of the stochastic game tug-of-war with noise and running payoff. As a consequence, we obtain game-theoretic proofs for the same regularity properties for…

In this paper we prove existence of entropy solutions to the time-fractional porous medium type equation, $$\partial_t[k\ast(u-u_0)]-\operatorname{div} (A(t,x)\nabla\varphi(u))=f\text{ in }Q_T=(0,T)\times\Omega,$$ with Dirichlet boundary…

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=\Delta u +f(x,t,u,\nabla u) &…

This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial…