Hi I have background on ordinary differential equations of the form $$u'=f(t,u).$$
I am interested in using what I know about ODE to learn aspect of partial differential equations of the form
$$u_t=\Delta u+f(t,u)$$ with Dirichlet/Neuman boundary conditions.
I am used to ODE, so when I see a problem of the form $u_t=\Delta u+f(t,u)$, I see an additional challenge which is the space variable $x$ and the Laplacian. The way I usually see people deal with this challenge is to multiply the PDE by $u$ then integrating over the domain $\Omega$ so they obtain a differential inequality with only a time variable.
I am looking for a book that deals with equations of the form $u_t=\Delta u+f(t,u)$, specially:
- existence of solution on $[0,\infty )$ (global existence)
- positive solutions ($u_0>0$ implies $u(t,x)>0$ for all $t,x$)
- asymptotic behaviour