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Let $U\subset\mathbb{R}^{n}$ be open, $U_{T}$ and $\Gamma_{T}$ be the parabolic cylinder and boundary of $U$ for arbitrary $0\leq t\leq T$, respectively, and suppose $u$ solves $$ \left\{\begin{array}{rl} u_{t}-\Delta u=u^{\alpha}&\text{in}\;U_{T}\\ u=g&\text{on}\;\Gamma_{T}\end{array}\right. $$ where $0\leq g\leq1$. If $u\in\mathscr{C}^{2,1}(U_{T})\cap\mathscr{C}(U_{T}\cup\Gamma_{T})$, then determine which values of $\alpha$ yield uniformly bounded solutions for all $T>0$.

First of all, I just want to say I am only starting to learn about PDEs from Evans' text, and am only on Chapter 2 of Part I, which deals with the usual linear second order PDE. I suspect the restriction of initial/boundary conditions to some $g$ with $0\leq g\leq1$ is probably fairly significant in simplifying the problem. But even for this case, I am not sure how to attack the problem. But is it reasonable to expect that one should know how to analyze the solutions to this PDE for arbitrary $g$?

Some of the ways I tried to attack the problem included finding a simplifying substitution (for example, when $\alpha=1$ I was able to use $u=e^{t}v$ to eliminate the RHS term), but I could not think of any that would work for all $\alpha$. Next I tried applying usual energy techniques, but this did not seem to lead anywhere fruitful (things like the arbitrariness of $\alpha$ and the non-periodic boundary conditions obstructed a clear path to making use any possible candidate for the energy functional).

Next I wondered if solutions to the PDE itself (without any specified boundary conditions) satisfies a maximum principle. But I found this hard to prove using straightforward techniques like $\epsilon$ regularization, and the methods Evans uses to prove the strong maximum principle for just the heat equation via time-dependent mean-value properties was already complicated enough as it was to even think about extending the arguments to the non-linear PDE above. On the other hand, the non-linearity likely precludes this possibility, at least for all $\alpha$. Furthermore, if for all $\alpha$ the equation did satisfy a weak maximum principle, then it would imply the answer is all $\alpha\in\mathbb{R}$, and this also seems unlikely. Why it seems unlikely is related to my next thought in considering the ODE $$u_{t}=u^{\alpha}.$$ For $t>0$ and $0\leq u(0)=g(\cdot,0)\leq1$, solutions to this ODE exhibit a wide variety of behavior depending on $\alpha$, and even lacks uniqueness in many cases. But in general, either the solution blows up (e.g. $\alpha\geq2$), grows exponentially (e.g. $\alpha=1$), or else grows in some sort of algebraic fashion like $O(x^{\beta})$ (correct me if I am wrong, please!).

The point of this breakdown into various classes of solutions to the above ODE is to somehow relate the behavior back to the behavior of solutions to the PDE with the diffusion term $\Delta u$ added back. Heuristically, the diffusion term $\Delta u$ "smoothes" the solution and "spreads" it out over the spatial domain $U$ as $t>0$ progreses. On the other hand, $u^{\alpha}$ heuristically causes growth in the solution in space as $t>0$ evolves. If the growth term $u^{\alpha}$ overpowers the diffusion term $\Delta u$, then it seems the solution would not remain uniformly bounded in $t$, and if the growth term is very strong, then singularities could develop in finite time $t$. Anyway, that is all fine (I think), but the practical question is how does one convert heuristics about the PDE into specific quantitative information/estimates about the solution (in this case, uniform $t$-boundedness of $u$ as a function of $\alpha$)?

One last comment. I haven't really considered (from a purely heuristic point of view) the boundary terms, and the special assumption $0\leq g\leq1$ (I would still like to know in any answer how we could generalize $g$ a bit more, if it can be reasonably done). $U$ is of course bounded, so the diffusion of the system is constrained to a bounded region kept at some constant positive density on the boundary. This would seem to me that the growth term inevitably dominates, since the solution will continuously grow and grow (as mass is added into the system restricted to a bounded region). Whether the solution develops singularities would depend on the strength of the diffusion term. If the diffusion term is strong enough, then the mass added to the system has enough time to spread out, whereas if it is too weak, then the concentration will spike into a singularity at some point(s) in $U$. Either way, it would appear that unless $g(x,0)\equiv0$, the solution is at best not uniformly bounded, and at worst actually blows up in finite time $t<\infty.$ Maybe this answers the question? If it helps, I was only asked to "explain" my answer, so given the non-linearity of the PDE itself, perhaps we aren't expected to provide a rigorous argument, though I would like to develop one if it is possible.

Any hints are appreciated -- thanks!

EDIT 1: I should clarify since I don't know if the terminology parabolic boundary/cylinder is standard. $U$ is an open subset of $\mathbb{R}^{n}$ (spatial region) and $$U_{T}:=U\times(0,T],$$ $$\Gamma_{T}:=\partial U\times[0,T]\cup U\times\{t=0\}=\overline{U_{T}}-U_{T}.$$

EDIT 2: For problems addressing behavior, such as boundedness, for large $t$, is it helpful to consider the time-independent case? e.g. the elliptic PDE $$-\Delta u=u^{\alpha}.$$ In other words, if the solution to this elliptic PDE (with appropriate boundary conditions as intimated by the original parabolic-type PDE) is bounded, could we somehow deduce boundedness (and hence uniform boundedness) for all $t$ before "$t=\infty$?" I guess the only obstruction to this would be the possibility of singularities developing and then evaporating over time, which doesn't seem likely (though I am not going to say it's impossible, since I have no mathematical justification for it, and also black holes can be thought of as "singularities" which evaporate over large periods of time as they radiate energy, this probably going way off topic).

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Since you are reading the book by Evans anyway, you may want to consult section 9.4: blow up for $u_t=\Delta u+u^2$ with large data, and non-existence of solutions of $-\Delta u=u^\alpha$ for large $\alpha$. (The section number is for the 1st edition.) This section does not answer your question, but is certainly relevant. –  user53153 Feb 26 '13 at 0:16
    
For $\alpha=2$ Evans requires the boundary data to be $0$, and for arbitrary $\alpha$ he only considers the case where $U=\mathbb{R}^{n}$. He also relies on some things from Chapter 6 (eigenvalues of the operator $-\Delta$) which I am not yet familiar with. I will definitely read into it more thoroughly though - at the very least it will hopefully help indicate which $\alpha$ lead to solutions defined for all $T>0$ which are also uniformly bounded. Thanks for the suggestion! –  Taylor Martin Feb 26 '13 at 0:32
    
I'm pretty sure that the solution is monotone with respect to boundary (and initial) data: the bigger $g$, the bigger the solution. (This is not obvious due to nonlinearity, but should be true.) Then the worst case is $g\equiv 1$, and the best case is $g=0$ when of course $u\equiv 0$. –  user53153 Feb 26 '13 at 0:36
    
Let me just add to the reason why the black hole is way off topic that in classical physics (which is described by PDE's) black holes do not evaporate. –  timur Mar 4 '13 at 2:59
    
I think your way of approaching the problem by drawing analogy to ODEs is a good one. Make a conjecture on the threshold value of $\alpha$, and try to use maximum principles to show the the solutions stay bounded if $\alpha$ is below that threshold. Then try to construct an example that blows up when $\alpha$ is above that threshold. You might need maximum/comparison principles also to show blow up. –  timur Mar 4 '13 at 3:03

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