arXiv subjects

Shaomei Fang

Publications and source records attributed to Shaomei Fang.

5 recordsLinked to original sources

arXiv2024

Asymptotic behavior for the fast diffusion equation with absorption and singularity

This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of $u_t=\triangle u^m -u^p$. We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies $0 1$. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\frac{n-1}{n} m+\frac{2}{n}$ via the entropy dissipation method combining the generalized Shannon's inequality and Csisz$\mathrm{\acute{a}}$r-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.

math.AP↗

arXiv2017

Large time behavior of solutions for a Cauchy problem on nonlinear conservation laws with large initial data in the whole space

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, frequency decomposition, pseudo-differential operators, we obtain the global existence and the optimal decay estimate of $t$.

math.AP↗

arXiv2017

The Brezis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-Δ)^{s} u-μ\frac{u}{|x|^{2s}}-λu=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} Ω,\ \ \, u=0\,&\text{in} \R^N\backslashΩ \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Brézis-Nirenberg type result when $f(x,u)=0$.

math.AP↗

arXiv2017

The global existence and time-decay for the solution of the fractional pseudo-parabolic equation

We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space $R^n,n\geq 1$. Here, the fractional order $α$ is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for $0<α< 1$ and $α> 1$, respectively. We establish the global existence and time-decay rates for small-amplitude classical solutions to the Cauchy problem for $α>0$. In the case that $0<α< 1$ , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.

math.AP↗

arXiv2017

A Global Compact Result for a Fractional Elliptic Problem with Critical Sobolev-Hardy Nonlinearities on ${\mathbb R}^N$

In this paper, we are concerned with the following type of elliptic problems: $$ (-Δ)^α u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u, u\,\in\,H^α({\mathbb R}^N), $$ where $2<q< 2^*$, $0<α<1$, $0<s<2α$, $2^*_{s}=2(N-s)/(N-2α)$ is the critical Sobolev-Hardy exponent, $2^*=2N/(N-2α)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to the problem, we obtain the existence of positive solutions under certain assumptions on $a(x),k(x)$.

math.AP↗