Convergence of the Dirichlet solutions of the very fast diffusion equation

Abstract

For any $-1<m<0$, $\mu >0$, $0\le u_0\in L^{\infty }(R)$ such that $u_0(x)\le ({\mu }_0|m||x|{)}^{\frac{1}{m}}$ for any $|x|\ge R_0$ and some constants $R_0>1$ and $0<{\mu }_0\le \mu$, and $f,g\in C([0,\infty ))$ such that $f(t),g(t)\ge {\mu }_0$ on $[0,\infty )$ we prove that as $R\to \infty$ the solution $u^R$ of the Dirichlet problem $u_t=(u^m/m{)}_{xx}$ in $(-R,R)\times (0,\infty )$, $u(R,t)=(f(t)|m|R{)}^{1/m}$, $u(-R,t)=(g(t)|m|R{)}^{1/m}$ for all $t>0$, $u(x,0)=u_0(x)$ in $(-R,R)$, converges uniformly on every compact subsets of $R\times (0,T)$ to the solution of the equation $u_t=(u^m/m{)}_{xx}$ in $R\times (0,\infty )$, $u(x,0)=u_0(x)$ in $R$, which satisfies ${\int }_{R}u(x,t)dx={\int }_R{u}_{0}dx-{\int }_0^t(f(s)+g(s))ds$ for all $0<t<T$ where ${\int }_0^T(f+g)ds={\int }_R{u}_{0}dx$. We also prove that the solution constructed is equal to the solution constructed in [Hu3] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.

33 pages