### A discontinuous problem involving the $p$-Laplacian operator and critical exponent in ${\mathbb{R}}^{N}$.

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We consider the Cahn-Hilliard equation in ${H}^{1}\left({\mathbb{R}}^{N}\right)$ with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as $\left|u\right|\to \infty $ and logistic type nonlinearities. In both situations we prove the ${H}^{2}\left({\mathbb{R}}^{N}\right)$-bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012).

The Cauchy problem for a semilinear abstract parabolic equation is considered in a fractional power scale associated with a sectorial operator appearing in the linear main part of the equation. Existence of local solutions is proved for non-Lipschitz nonlinearities satisfying a certain critical growth condition.

We show that the critical nonlinear elliptic Neumann problem $\Delta u-\mu u+{u}^{7/3}=0$ in $\Omega $, $u>0$ in $\Omega $, $\frac{\partial u}{\partial \nu}=0$ on $\partial \Omega $, where $\Omega $ is a bounded and smooth domain in ${\mathbb{R}}^{5}$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu}_{K}>0$ such that for $0<\mu <{\mu}_{K}$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega $, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

In questa nota dimostriamo stime asintotiche ottimali per le soluzioni deboli non negative del problema al contorno $$-{\mathrm{\Delta}}_{{\mathbb{H}}^{n}}u={u}^{\left(Q+2\right)/\left(Q-2\right)}in\mathrm{\Omega},u=0\text{in}\partial \mathrm{\Omega}.$$$-{\mathrm{\Delta}}_{{\mathbb{H}}^{n}}$

We study the leading order behaviour of positive solutions of the equation $-\Delta u+\u03f5u-{\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u=0,\phantom{\rule{2.0em}{0ex}}x\in {\mathbb{R}}^{N}$, where $N\ge 3$, $q>p>2$ and when $\u03f5>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent ${2}^{*}=\frac{2N}{N-2}$. For $p<{2}^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>{2}^{*}$ the solution asymptotically coincides...

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u\u209c-\Delta \left({a}_{11}u\right)=h(t,x){\left|v\right|}^{p}$, $v\u209c-\Delta \left({a}_{21}u\right)-\Delta \left({a}_{22}v\right)=k(t,x){\left|w\right|}^{q}$, $w\u209c-\Delta \left({a}_{31}u\right)-\Delta \left({a}_{32}v\right)-\Delta \left({a}_{33}w\right)=l(t,x){\left|u\right|}^{r}$, for $x\in {\mathbb{R}}^{N}$, t > 0, p > 0, q > 0, r > 0, ${a}_{ij}={a}_{ij}(t,x,u,v)$, under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x\in {\mathbb{R}}^{N}$, where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the above system,...

This paper is mainly concerned with the blow-up and global existence profile for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources.

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.

We focus on the blow-up in finite time of weak solutions to the wave equation with interior and boundary nonlinear sources and dissipations. Our central interest is the relationship of the sources and damping terms to the behavior of solutions. We prove that under specific conditions relating the sources and the dissipations (namely p > m and k > m), weak solutions blow up in finite time.

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...

In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]