arXiv subjects
Publications and source records attributed to Shaopeng Xu.
In this paper, we consider the following problem: \[ \begin{cases} -\nabla\cdot A(x,u,\nabla u) + H(x,u,\nabla u) = f(x), & x \in \Omega, u = 0, & x \in \partial \Omega, \end{cases} \] in a bounded open set \( \Omega \subset \mathbb{R}^N \). We have established certain gradient estimates and proved the existence of a renormalized solution for the equation.
In this paper, we consider the following nonlinear parabolic equation with non-coercive terms in \(R^N\) space \[ \dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ \Phi(x,t,\nabla u))=f, \text{ in }\Omega \times (0,T). \] Here \(\Omega\) is a bounded open set of \(R^N\) with the boundary \(\partial \Omega\) satisfying Lipschitz condition. The Carath\'eodory function \(\Phi\) is restricted by $|\Phi(x,t,s)|\le c(x,t)|s|^\gamma$ with parameters depending on $p$ and $N$. And the initial value $u(x,0)=u_0(x)$. For convenience, we define the domain $Q := \Omega \times (0,T)$ and the boundary similarly. Then for $f\in L^1(Q)$ and $u_0\in L^1(\Omega)$, we prove the existence and uniqueness of a renormalized solution via truncation methods, monotone operator theory, and a prior gradient estimates.
We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and \[ \left\lvert H(x,t,\nabla u)\right\rvert\leqslant g(x,t)\left\lvert \nabla u\right\rvert^{\delta(x)} \] with \[ \delta (x)=\frac{p(x)(N+1)-N}{(N+2)(p(x)-1)}(p^--1) \quad \text{and} \quad p^-=\underset{x\in\bar{\Omega}}{min}\,p(x). \] By virtue of truncation metheod,the monotone operator theory and a gradient estimate we prove existence of renormalized solutions without coercivity condition on lower order term in the framework of variable exponents.
\indent In this paper, we study a class of parabolic-elliptic Keller-Segel systems with diffusion sensitivity dependent on spatial position, given by type \begin{equation} \left\{ \begin{array}{ll} u_{t} = \bigtriangledown\cdot(|x|^{\beta} \bigtriangledown u)-\bigtriangledown\cdot(u^{\alpha} \bigtriangledown v), 0=\bigtriangleup v-\mu +u, \qquad \mu:=\frac{1}{|\Omega|}\int_{\Omega}udx,\end{array}\right. \end{equation} under homogeneous Neumann conditions in a ball $\Omega=B_{R}(0)\subset \mathbb{R}^{n}$ with $\alpha \ge 1$, $\beta>0$ and $n\ge 2$.\par \indent It is proved that any nonconstant nonnegative radial initial data $u_{0}\in C^{\theta}(\overline{\Omega})$, where $\theta \in (0,1)$, there exists a radially symmetric classical solution of the system (0.1) in $(\Omega \setminus \{ 0 \})\times (0,T)$ for some $T>0$; moreover, if the initial values $u_{0}\in C^{1+\theta}(\overline{\Omega})$ for some $\theta \in (0,1)$ and satisfy a certain compatibility criterion and are radially decreasing, then this solution is bounded and unique in $(\Omega \setminus \{ 0 \})\times (0,T^{*})$ with $T^{*}<T$.\par Finally, it is found that the initial mass corresponding to this parabolic-elliptic problem (0.1) is sufficiently concentrated to allow the solution to blow up in finite time.
We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (\phi(u)\nabla u) - \nabla \cdot (\psi(u)\nabla v)+f(u), & x \in \Omega, \; t > 0, v_t = \Delta v - v + u, & x \in \Omega, \; t > 0, \end{cases} \end{align*} in a smooth bounded domain \(\Omega \subset \mathbb{R}^n\) with \(n \geq 2\), the Neumann initial-boundary value problem admits a globally defined, uniformly bounded classic solution for all sufficiently regular non-negative initial data \(u_0\) and \(v_0\). In the first equation, assume that \(\phi\) and \(\psi\) are dominated by a logarithmic function and a polynomial respectively. The logical source \(f\) representing the natural growth and decay of cells satisfies \(f \in W^{1,\infty}_{\mathrm{loc}}(\Omega)\) and \(f(0) \geq 0\). Then we will see that the unique solution \(u \in C^{2,1}((\overline{\Omega}) \times [0,T] )\) and \(v \in W^{1,q}([0,T] ; C^{2,1}(\overline{\Omega}))\).