Related papers: Quasilinear generalized parabolic Anderson model e…

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures. This work deals with…

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider…

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=\Delta u(t,z)+\xi(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(\xi(z):z\in{\mathbb{Z}}^d)$. We consider independent and…

Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225-2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483-533], this work is concerned with the precise spatial asymptotic…

We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=\Delta u(t,x) +\xi_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the $\xi$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and…

We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equations, solutions of which require renormalization and…

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…

We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear…

We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic H\"older spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.

We show that the random field solution to the parabolic Anderson equation $(\partial_t-\frac12 \Delta)u=u\diamond \dot{W}$ is jointly H\"older continuous in space and time.

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the…

The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an…

In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on the kind of problems arising for example when controlling the anisotropic Allen-Cahn equation as a model for crystal growth.…

It is known that the Swift-Hohenberg equation $\partial u/\partial t = -(\partial_x^2 + 1)^2u + \varepsilon (u-u^3)$ can be reduced to the Ginzburg-Landau equation (amplitude equation) $\partial A/\partial t = 4\partial_x^2 A + \varepsilon…

In this paper, we consider the following quasilinear Schr\"{o}dinger equation \begin{align*} -\Delta u-u\Delta(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,}…

We show that the parabolic equation $u_t + (-\Delta)^s u = q(x) |u|^{\alpha-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times…

We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show…

The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…

We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-\Delta u+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp(\frac{\omega(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $\omega$ a positive…