We present some general results on minimal barriers in the sense of De Giorgi for geometric evolution problems. We also compare minimal barriers with viscosity solutions for fully nonlinear geometric problems of the form $u_t+F\left(t,x,\nabla u,{\nabla }^{2}u\right)=0$. If $F$ is not degenerate elliptic, it turns out that we obtain the same minimal barriers if we replace $F$ with $F^{+}$, which is defined as the smallest degenerate elliptic function above $F$.

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.

We investigate critical exponents for blow-up of nonnegative solutions to a class of parabolic inequalities. The proofs make use of a priori estimates of solutions combined with a simple scaling argument.

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 $C₁{(T-t)}^{-1/(p-1)}{\le \left|\right|u\left(t\right)\left|\right|}_{\infty }\le C₂{(T-t)}^{-1/(p-1)}$. 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 develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ in a smooth domain $\Omega$ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity...

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...

This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi \left(v\right)$ and the growth term $f\left(u\right)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0<\chi \left(v\right)\le {\chi }_0/v^k$$(k\ge 1...$