### Recent results and open problems on parabolic equations with gradient nonlinearities.

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We present some recent results on the blow-up behavior of solutions of heat equations with nonlocal nonlinearities. These results concern blow-up sets, rates and profiles. We then compare them with the corresponding results in the local case, and we show that the two types of problems exhibit "dual" blow-up behaviors.

Consider the nonlinear heat equation (E): ${u}_{t}-\Delta u={\left|u\right|}^{p-1}u+b{\left|\nabla u\right|}^{q}$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C\u2081{(T-t)}^{-1/(p-1)}{\le \left|\right|u\left(t\right)\left|\right|}_{\infty}\le C\u2082{(T-t)}^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality ${u}_{t}-{u}_{xx}\ge {u}^{p}$. More general inequalities of the form ${u}_{t}-{u}_{xx}\ge f\left(u\right)$ with, for instance, $f\left(u\right)=(1+u)lo{g}^{p}(1+u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary...

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$: either the space derivative ${u}_{x}$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in ${C}^{1}$ norm to the unique steady state. The main difficulty is to prove ${C}^{1}$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...

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