相关论文: The Fractional Stefan Problem: Global Regularity o…

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0,…

We study the Cauchy problem for the semilinear fractional heat equation $u_{t}=\triangle^{\alpha/2}u+f(u)$ with non-negative initial value $u_{0}\in L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For $f$…

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate…

In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of…

We consider time fractional stochastic heat type equation $$\partial^\beta_tu(t,x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$,…

Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is…

We study coupled non-linear parabolic equations for a fluid described by a material density and a temperature, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on…

The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical…

We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey…

We consider nonlinear diffusion equations of the form $\partial_t u= \Delta \phi(u)$ in $\mathbb R^N$ with $N \ge 2.$ When $\phi(s) \equiv s$, this is just the heat equation. Let $\Omega$ be a domain in $\mathbb R^N$, where $\partial\Omega$…

In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space one-phase Stefan problem in terms of the three parametric Mittag-Leffer function $E_{\alpha,m;l}(z)$. We consider Dirichlet and Newmann…

We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary…

The time-fractional diffusion equation is considered, where the time derivative is either of Caputo or Riemann-Liouville type. The solution of a general initial-boundary value problem with time-dependent boundary conditions over bounded and…

The one-dimensional (1D) Stefan problem is a prototypical heat and mass transfer problem that analyzes the temperature distribution in a material undergoing phase change. In addition, it describes the evolution of the phase change front…

We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with…

In this paper, we study the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-\Delta)^{\sigma/2} u(t,x)+b(t) \partial_t u(t,x)…

We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the $p$-Laplacian with non-zero right hand side, where $p\in (1,\infty)$. This is the optimal regularity for the problem. We also obtain…

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional…

We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-\Delta)^s u = \chi_{\{u>c\}} \] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free…

We study the fluctuations of the outer domain of Hastings-Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process $\mathcal{F}$ taking values in the space of holomorphic functions on $\{…