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 study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.

We consider a special type of a one-dimensional quasilinear wave equation wtt - phi (wt / wx) wxx = 0 in a bounded domain with Dirichlet boundary conditions and show that classical solutions blow up in finite time even for small initial data in some norm.

Let $T$ be a positive number or $+\infty$. We characterize all subsets $M$ of ${ℝ}^n\times ]0,T[$ such that $$\underset{X\in {ℝ}^n\times ]0,T[}{inf}u\left(X\right)=\underset{X\in M}{inf}u\left(X\right)i$$ for every positive parabolic function $u$ on ${ℝ}^n\times ]0,T[$ in terms of coparabolic (minimal) thinness of the set $M_{\delta }={\cup }_{(x,t)\in M}{B}^p((x,t),\delta t)$, where $\delta \in (0,1)$ and $B^p((x,t),r)$ is the “heat ball” with the “center” $(x,t)$ and radius $r$. Examples of different types of sets which can be used instead of “heat balls” are given. It is proved that (i) is equivalent to the condition ${sup}_{X\in {ℝ}^n\times {ℝ}^{+}}u\left(X\right)={sup}_{X\in M}u\left(X\right)$ for every bounded parabolic function on ${ℝ}^n\times {ℝ}^{+}$ and hence to all equivalent conditions given in the article [7]....