arXiv subjects
Publications and source records attributed to Martino Prizzi.
We consider a semilinear wave equation in the whole space with a deep potential well. We prove that as the depth of the well tends to infinity, the solutions of the equation converge to the solutions of a wave equation defined on the bottom of the well, with Dirichlet condition on the boundary.
We consider a parabolic equation whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. Combining a recent conditional stability result with a well posed variational problem, we reconstruct the initial condition of an unknown solution from a rough measurement at the final time.
We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jaeh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hoelder and logarithmic), arbitrarily closed, but not up to the final time.
We prove some $C^\infty$ and Gevrey well-posedness results for hyperbolic equations whose coefficients lose regularity at one point.
We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgodd continuous in time. This result fills the gap between uniqueness and continuity results obtained so far.
The interest of the scientific community for the existence, uniqueness and stability of solutions to PDE's is testified by the numerous works available in the literature. In particular, in some recent publications on the subject an inequality guaranteeing stability is shown to hold provided that the coefficients of the principal part of the differential operator are Log-Lipschitz continuous. Herein this result is improved along two directions. First, we describe how to construct an operator, whose coefficients in the principal part are not Log-Lipschitz continuous, for which the above mentioned inequality does not hold. Second, we show that the stability of the solution is guaranteed, in a suitable functional space, if the coefficients of the principal part are Osgood continuous.
Using Bony's paramultiplication we improve a result obtained in in a previous paper for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ regularity is replaced by Lipschitz regularity in $x$.
Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation u_{tt}+αu_t+β(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\timesΩ, u = 0, (t,x)\in[0,+\infty[\times\partialΩin $H^1_0(Ω)\times L^2(Ω) has finite Hausdorff and fractal dimension. Here $Ω$ is a regular, possibly unbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff and fractal dimension of $\mathcal I$ in terms of the structure parameters of the equation.
Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation u_t+β(x)u-Δu&=f(x,u),&&(t,x)\in[0,+\infty[\timesΩ, u&=0,&&(t,x)\in[0,+\infty[\times\partialΩ} {equation*} in $H^1_0(Ω)$ has finite Hausdorff dimension. Here $Ω$ is an arbitrary, possibly unbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of subcritical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an an attractor. If $Ω$ is regular, $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff dimension of $\mathcal I$ in terms of the structure parameter of the equation.
Under fairly general assumptions, we prove that every compact invariant subset $\mathcal I$ of the semiflow generated by the semilinear damped wave equation εu_{tt}+u_t+β(x)u-\sum_{ij}(a_{ij} (x)u_{x_j})_{x_i}&=f(x,u),&& (t,x)\in[0,+\infty[\timesΩ, u&=0,&&(t,x)\in[0,+\infty[\times\partialΩin $H^1_0(Ω)\times L^2(Ω)$ is in fact bounded in $D(\mathbf A)\times H^1_0(Ω)$. Here $Ω$ is an arbitrary, possibly unbounded, domain in $\R^3$, $\mathbf A u=β(x)u-\sum_{ij}(a_{ij}(x)u_{x_j})_{x_i}$ is a positive selfadjoint elliptic operator and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an an attractor.
We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.
We prove existence of global attractors for damped hyperbolic equations of the form $$\aligned \eps u_{tt}+α(x) u_t+β(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u),\quad x\in Ω, t\in[0,\infty[, u(x,t)&=0,\quad x\in \partial Ω, t\in[,\infty[.\endaligned$$ on an unbounded domain $Ω$, without smoothness assumptions on $β(\cdot)$, $a_{ij}(\cdot)$, $f(\cdot,u)$ and $\partialΩ$, and $f(x,\cdot)$ having critical or subcritical growth.
We give sharp regularity conditions, ensuring the backward uniquess property to a class of parabolic operators.
We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitely computed and can be exploited to detect the existence of recurrent dynamics in the equation.
We consider the parabolic equation $$u_t-Δu=F(x,u),\quad (t,x)\in\R_+\times\R^n\tag{P}$$ and the corresponding semiflow $π$ in the phase space $H^1$. We give conditions on the nonlinearity $F(x,u)$, ensuring that all bounded sets of $H^1$ are $π$-admissibile in the sense of Rybakowski. If $F(x,u)$ is asymptotically linear, under appropriate non-resonance conditions, we use Conley's index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained in this paper extend earlier results of Rybakowski concerning parabolic equations on {\it bounded} open subsets of $\R^n$.
We prove some $C^\infty$ and Gevrey well-posedness results for hyperbolic equations with singular coefficients.
In this paper we study a family of semilinear reaction-diffusion equations on thin spatial domains, lying close to a lower dimensional submanifold $M$. As the thickness tends to zero, the domains collapse onto (a subset of) $M$. As it was proved in a previous paper (M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Stud. Math. 151), the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain abstract limit phase space. One of the objectives of this paper is to give more manageable characterizations of the limit phase space. Under additional hypotheses, we also give a simple description of the limit equation. If, in addition, $M$ is a sphere and the nonlinearity of the above equations is dissipative, we prove that, if the thickness is small enough, the corresponding equation possesses an inertial manifold, i.e. an invariant manifold containing the attractor of the equation. We thus obtain the existence of inertial manifolds for reaction-diffusion equations on certain classes of thin domains of genuinely high dimension.
We prove the existence of a compact L^2-H^1 attractor for a reaction-diffusion equation in R^n. This improves a previous result of B. Wang concerning the existence of a compact L^2-L^2 attractor for the same equation.