Critical exponent and blow-up rate for the omega-diffusion equations on graphs with Dirichlet boundary conditions
Weican Zhou, Miaomiao Chen, Wenjun Liu · DOAJ (DOAJ: Directory of Open Access Journals) · 2014
In this article, we study the $\omega$-diffusion equation on a graph with Dirichlet boundary conditions $$\displaylines{ u_t(x,t)=\Delta_{\omega}u(x,t)+e^{\beta t}u^{p}(x,t), \quad (x,t)\in S\times(0,\infty), \cr u(x,t)=0, \quad (x,t)\in \partial S\times[0,\infty), \cr u(x,0)=u_0(x)\geq0, \quad x\in V, }$$ where $\Delta_{\omega}$ is the discrete weighted Laplacian operator. First, we prove the existence and uniqueness of the local solution via Banach fixed point theorem. Then, by the method of supersolutions and subsolutions we prove that the $\omega$-diffusion problem has a critical exponent $p_{\beta}$: when $p>p_{\beta}$, the solution becomes global; while when $1<p