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...

We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧$\partial (e^{\beta u}-1)/\partial t-{div\left(\right|\nabla u|}^{p-2}\nabla u)+div(c(x,t){\left|u\right|}^{s-1}u)+b(x,t){\left|\nabla u\right|}^r=f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{\beta u}-1)(x,0)=(e^{\beta u₀}-1)\left(x\right)$ in Ω. with s = (N+2)/(N+p) (p-1), $c(x,t)\in {\left(L^{\tau }\left(QT\right)\right)}^N$, τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b(x,t)\in L^{N+2,1}\left(QT\right)$ and f ∈ L¹(Q).

An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on $X^{\alpha }$. This semigroup possesses an $(X^{\alpha }-Z)$-global attractor that is closed, bounded, invariant in $X^{\alpha }$, and attracts bounded subsets of $X^{\alpha }$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...

Let $u\in L^2(I;H^1\left(\Omega \right))$ with ${\partial }_{t}u\in L^2(I;H^1{\left(\Omega \right)}^{*})$ be given. Then we show by means of a counter-example that the positive part $u^{+}$ of $u$ has less regularity, in particular it holds ${\partial }_t{u}^{+}\notin L^1(I;H^1{\left(\Omega \right)}^{*})$ in general. Nevertheless, $u^{+}$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

We consider $2m$th-order $(m\ge 2)$ semilinear parabolic equations $u_t=-{\left(-\Delta \right)}^{m}u\pm {\left|u\right|}^{p-1}u$ in ${ℛ}^N\times {ℝ}_{+}$ $(p>1)$, with Dirac’s mass $\delta \left(x\right)$ as the initial function. We show that for $p<p_0=1+2m/N$, the Cauchy problem admits a solution $u(x,t)$ which is bounded and smooth for small $t>0$, while for $p\ge p_0$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ${\partial }_{t}u+{div}_{y}A(y,u)-{\Delta }_{y}u=0$, where $y\in {ℝ}^N$ and the flux $A$ is periodic in $y$. More specifically, we consider the case when the initial data is an $L^1$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^1$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first...

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi \nabla ·(u\nabla v/v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega \subset {ℝ}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ which is bounded when $\chi >0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.

We consider positive solutions of the system $u_t-\Delta u=v^p$; $v_t-\Delta v=u^q$ in a ball or in the whole space, with $p,q>1$. Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p=q$. Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...

Sfruttando i risultati di [1], si prova che le derivate spaziali $D^{\alpha }u$ di ordine $|\alpha |$ con delle soluzioni in $Q$ di un sistema parabolico quasilineare di ordine $2m$ con andamenti strettamente controllati, sono parzialmente hölderiane in $Q$ con esponente di hölderianità decrescente al crescere di $|\alpha |$.

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺. ...

We consider a nonlinear parabolic system modelling chemotaxis $u_t=\nabla ·(\nabla u-u\nabla v)$, $v_t=\Delta v+u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in ${ℝ}^n$. The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta }$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta$. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator...

Let $X={\sum }_{i=1}^k{a}_i{U}_i$, $Y={\sum }_{i=1}^k{b}_i{U}_i$, where the $U_i$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_i,b_i$ are real constants. We prove that if $b{²}_i$ is majorized by $a{²}_i$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^p$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

In this paper we examine self-similar solutions to the system $u_{it}-d_i\Delta u_i={\prod }_{k=1}^m{u}_k^{p_k^i}$, i = 1,…,m, $x\in {ℝ}^N$, t > 0, $u_i(0,x)=u_{0i}\left(x\right)$, i = 1,…,m, $x\in {ℝ}^N$, where m > 1 and $p_k^i>0$, to describe asymptotics near the blow up point.

Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti dell’equazione di evoluzione non autonoma $u^{\prime }(t)\mathrm{—}A(t)u(t)=f(t)$, con il dato iniziale $u(0)=x$, in spazi di Banach. I dominii degli operatori $A(t)$ variano in $t$ e non sono necessariamente densi in $E$. Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità holderiana della soluzione e della sua derivata.

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we...

Sfruttando i risultati di [1], si prova che le derivate spaziali $D^{\alpha }u$ di ordine $|\alpha |$ con delle soluzioni in $Q$ di un sistema parabolico quasilineare di ordine $2m$ con andamenti strettamente controllati, sono parzialmente hölderiane in $Q$ con esponente di hölderianità decrescente al crescere di $|\alpha |$.

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $uₜ-\Delta \left(a_{11}u\right)=h(t,x){\left|v\right|}^p$, $vₜ-\Delta \left(a_{21}u\right)-\Delta \left(a_{22}v\right)=k(t,x){\left|w\right|}^q$, $wₜ-\Delta \left(a_{31}u\right)-\Delta \left(a_{32}v\right)-\Delta \left(a_{33}w\right)=l(t,x){\left|u\right|}^r$, for $x\in {ℝ}^N$, 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 $x\in {ℝ}^N$, 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...

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some...

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\mathrm{\Omega }$ made of two materials. The first one is located in a set $F_{ϵ}$ of cylindrical parallel fibers periodically distributed with a period of size $ϵ$, and the second one is located in the "matrix" $M_{ϵ}=\mathrm{\Omega }\setminus F_{ϵ}$. The ratio between the conductivity coefficients of the two materials is of order $1/{ϵ}^2$. After writing the homogenized problem, we give a corrector result and prove...

We consider the systems of hyperbolic equations ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S1) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$ ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S2) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{l(t,x)\left|v\right|}^m+k(t,x){\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$, (S3) ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|u\right|}^p$, t > 0, $x\in {ℝ}^N$, ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|v\right|}^q$, t > 0, $x\in {ℝ}^N$, in $(0,\infty )\times {ℝ}^N$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.