I want to learn to use matlab to study the graphical behavior of parabolic partial differential equation solutions, in the sense of trying to simulate when the solution will be global or when it will blow-up. I have several questions, forgive me for the layman's questions if that's the case. For example, if I have the PDE
$ \begin{equation}\label{Fujita} \left\{ \begin{array}{rll} u_t - \Delta u &= u^{p}& \mbox{ in } \mathbb{R}^n \times (0,T), \\ {u}(0) &= {u}_{0}& \mbox{ in } \mathbb{R}^n \\ \end{array} \right. \end{equation}$
it is known that for $p<1+\dfrac{2}{n}$ the solution blow-up in finite time(in the sense $\lim_{t \rightarrow T_{max}}||u(t)||_{\infty} = \infty$), and for $p>1+\dfrac{2}{n}$ exists a global solution. Is it possible to simulate this in MATLAB? I believe he cannot do it in unlimited domains, but is it possible to do something?
Another question, in the case of the solution to the equation above, I have a classic solution to the problem. But for example, if instead of $u^p$ I had $|x|^{\gamma}u^p$, with $\gamma<0$ there is no classical solution, only Mild Solutions, what are solutions of the integral equation $$ \begin{equation}\label{mild-sol} \mathcal{F}u(t) = S(t)u_0 + \int_{0}^{t} S(t-s)|x|^{\gamma}u^p(s) ds \end{equation}$$
Here for every $t > 0$, $S(t)$ denotes the heat semigroup $$[S(t)f](x) := \int_{\mathbb{R}^n} K(x,y;t)f(y)dy,$$ where $K(x,y,t)$ is the heat Kernel of heat defined by $$K(x,y;t) = (4\pi t)^{-\frac{n}{2}}e^{-\frac{|x^2|}{4t}}, \quad t >0, \quad x\in \mathbb{R}^n.$$ Question: Is there a way to simulate the behavior Mild solution in MatLab? Is there any numerical formulation?
Thanks for any clarifications or suggestions for books to study, as I'm really at a loss to know what can and can't be done with the software.