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White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions…
We study the first vanishing time for solutions of the Cauchy-Dirichlet problem to the semilinear $2m$-order ($m \geq 1$) parabolic equation $u_t+Lu+a(x) |u|^{q-1}u=0$, $0<q<1$ with $a(x) \geq 0$ bounded in the bounded domain $\Omega…
We study the Euler-Korteweg equations with a weak capillarity tensor. It formally converges to the Euler equations in the zero capillarity limit. Our aim is two-fold : first we prove rigorously this limit in R d , d $\ge$ 1, and obtain a…
In this paper we study the Zakharov system on the upper half--plane $U=\{(x ,y)\in \R^2: y>0\}$ with non-homogenous boundary conditions. In particular we obtain low regularity local well--posedness using the restricted norm method of…
We consider numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational…
We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…
In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation $\ep^2\div\left(a(x)\nabla u\right)-a(x)F'(u)=0$ in $\R^2$, where $\ep\to 0$. The nodal set of these solutions is…
We develop a new infinite dimensional gluing method for fractional elliptic equations. As a model problem, we construct solutions of the fractional Allen--Cahn equation vanishing on a rotationally symmetric surface which resembles a…
We consider the $2 \times 2$ parabolic systems \begin{equation*} u^{\epsilon}_t + A(u^{\epsilon}) u^{\epsilon}_x = \epsilon u^{\epsilon}_{xx} \end{equation*} on a domain $(t, x) \in ]0, + \infty[ \times ]0, l[$ with Dirichlet boundary…
In this paper, we consider a class of optimal control problems for a one-dimensional time-discrete constrained quasilinear diffusion state-systems of singular Allen--Cahn types and its regularized approximating problems. We note that the…
In this paper we deal with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential…
In this paper, we study the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u + 2uh'(|u|^2)\Delta h(|u|^2) + F(|u|^2)u\mp A[h(|u|^2]^{2^*-1} h'(|u|^2)u,\ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x), \quad x\in…
For elliptic systems with block structure in the upper half-space and t-independent coefficients, we settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal…
In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-\Delta u=\lambda f(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\partial…
Let $u$ be a bounded positive solution to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is a locally Lipschitz continuous function. Among other things, we show that if…
This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a nontrivial…
We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u|^p$ in exterior domain in $\mathbf{R}^N$ $(N\geq 3)$. Especially, the lifespan of blowup solutions with small initial data are studied. The…
We study a mixed boundary value problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u(x))=0$ in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and…
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of…
After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we discuss what is known and the difference between the two-dimensional and higher-dimensional cases. Turning our attention to the two-dimensional case, we…