相关论文: Sharp Boundary Growth Rate Estimate of the Singula…

We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma , \] where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the…

We analyze the behavior of families $(u_\alpha)_{\alpha>0}$ of solutions to the high-order critical equation $P_\alpha u_\alpha=\Delta_g^k u_\alpha +\hbox{lot}=|u_\alpha|^{2^\star-2}u_\alpha$ on a Riemannian manifold $M$, with a uniform…

We construct unbounded positive $C^2$-solutions of the equation $\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in ${\R}^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in ${\R}^n$, the conformal…

We establish an area growth estimate for solutions that are bounded from above of the Liouville equation $\Delta u+K e^{2u}=0$ with a positive pinched curvature $0<\lambda\leq K\leq\Lambda$. As an application, we provide a new proof of…

We prove the growth rate of global solutions of the equation $u_t=\Delta u-u^{-\nu}$ in $\R^n\times (0,\infty)$, $u(x,0)=u_0>0$ in $\R^n$, where $\nu>0$ is a constant. More precisely for any $0<u_0\in C(\R^n)$ satisfying…

We study the Lane-Emden system $$\begin{cases} -\Delta u=v^p,\quad u>0,\quad\text{in}~\Omega, -\Delta v=u^q,\quad v>0,\quad\text{in}~\Omega, u=v=0,\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth…

In this paper, we mainly study the critical points and critical zero points of solutions $u$ to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain $\Omega$ in $\mathbb{R}^2$.…

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(\omega,t)u_{x^ix^j}+f)dt + (\sigma^{ik}(\omega,t)u_{x^i}+g^k)dw^k_t,…

Three decades ago Heath found the integral form of the exact analytic growing mode solution of the linear density perturbation $\delta$ on sub-horizon scales including the cosmological constant or the curvature term. Recently, we obtained…

We consider the problem of the approximation of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as $x^\alpha$, with $\alpha>1$. We propose an (semi-explicit) exponential-Euler…

We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…

We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -\Delta_p u + g(u) |\nabla u|^p &= \lambda f(u) \quad &&\mbox{in} \;\; \Omega, \\ u &=…

In this paper we propose some approaches for finding of pointwise estimates of a solution of the Dirichlet boundary value problem $-\Delta u \pm |u|^{q-1} u = 0 $, $|u|=k$ when $|x|=d<1$ and $|u|=0$ when $|x|=1$ where $x\in \Omega = \{x|…

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega,…

We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^\gamma \Delta u = f, \end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero…

In this manuscript we establish local H\"older regularity estimates for bounded solutions of a certain class of doubly degenerate evolution PDEs. By making use of intrinsic scaling techniques and geometric tangential methods, we derive…

We study the homogeneous Dirichlet problem for the equation \[ u_t-\operatorname{div}\left((a(z)\vert \nabla u\vert ^{p(z)-2}+b(z)\vert \nabla u\vert ^{q(z)-2})\nabla u\right)=f\quad \text{in $Q_T=\Omega\times (0,T)$}, \] where…

We study the Cauchy problem for the $3D$ compressible Euler equations under an arbitrary equation of state with positive speed of sound, aside from that of a Chaplygin gas. For open sets of smooth initial data with non-trivial vorticity and…

We investigate the following Dirichlet problem with variable exponents: \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u=\lambda \alpha (x)\left\vert u\right\vert ^{\alpha (x)-2}u\left\vert v\right\vert ^{\beta…

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation $-\Delta u=|u|^{p-1} u$ with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution $u$…