Nonlinear Parabolic Differential Equations: global existence and blow-up of solutions

DOTTORATOL'argomento principale della tesi è lo studio dell'esistenza globale e del blow-up di soluzioni ad alcune equazioni differenziali paraboliche nonlineari. La tesi è suddivisa in tre parti in ciascuna delle quali si prende in considerazione una diversa equazione. Nella Parte I, viene analizzato un problema di Cauchy per una equazione dei mezzi porosi con densit'a variabile che dipende solo dallo spazio, e un termine di diffusione del tipo potenza: questa equazione rappresenta un modello matematico per l'evoluzione della temperatura del plasma. Utilizzando metodi di sotto- e soprasoluzioni, grazie anche al principio del confronto, si determina quando la soluzione del problema esiste globalmente in tempo e quando invece avviene blow-up in tempo finito. Nella seconda parte, Part II, si studia una classe di equazioni di reazione-diffusione definita su varietà Riemanniane complete, noncompatte e di volume infinito. Queste equazioni contengono nonlinearity di tipo potenza e una diffusione lenta del tipo mezzi porosi. Per il problema di Cauchy relativo a queste equazioni, si dimostra esistenza globale in tempo delle soluzioni per dati iniziali positivi e che siano appartenenti ad opportuni spazi $L^p$. Inoltre, per queste soluzioni, si dimostra che esse sono limitate per tutti i tempi e si propone una stima quantitativa sulla loro norma $L^{infty}$. I metodi utilizzati per le dimostrazioni sono funzionali e si basano principalmente sulla validità delle disuguaglianze di Sobolev e Poincaré. Infine, nella Part III, si studia la nonesistenza di soluzioni per una classe di equazioni differenziali paraboliche quasilineari con un potenziale, definite in domini limitati. In particolare, si mostra come il comportamento del potenziale vicino alla frontiera del dominio e la nonlinearity di tipo potenza influenzano la nonesistenza delle soluzioni.The main topic of this thesis concerns the study of global existence and blow-up of solutions to certain nonlinear parabolic differential equations. The thesis is divided into three parts where three different equations are considered. In Part I, we analyze the Cauchy problem for the porous medium equation with a variable density, which depends on the space variable, and a power-like reaction term: this is a mathematical model of a thermal evolution of a heated plasma. Depending on the rate of decaying at infinity of the density function, by comparison method and suitable sub- and supersolutions, we determine whether the solution exists globally in time or blows up in finite time. In Part II, we consider reaction-diffusion equations posed on complete, noncompact, Riemannian manifolds of infinite volume. Such equations contain power-type nonlinearity and slow diffusion of the porous medium type. For the Cauchy problem related to this equation we prove global existence for positive initial data belonging to suitable $L^p$ spaces, and that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L$^infty$ norm. The methods of proof are functional analytic in character, as they depend solely on the validity of the Sobolev and the Poincar'{e} inequalities. In Part III, we are concerned with nonexistence results for a class of quasilinear parabolic differential equations with a potential in bounded domains. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions.DIPARTIMENTO DI MATEMATICA34MARAZZINA, DANIELECORREGGI, MICHEL

Global existence and blow-up of solutions to the porous medium equation with reaction and singular coefficients

We study global in time existence versus blow-up in finite time of solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term posed in the one dimensional interval $(-R,R)$, $R>0$. Here the weight function is singular at the boundary of the domain $(-R,R)$, indeed it is such that $\rho(x)\sim (R-|x|)^{-q}$ as $|x|\to R$, with $q\ge0$. We show a different behavior of solutions depending on the three cases when $q>2$, $q=2$ and $q<2$.Comment: arXiv admin note: text overlap with arXiv:2007.12005, arXiv:1911.06043, arXiv:1911.0730

Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density

We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in the sense that $\rho(x)\sim |x|^{-q}$ as $|x|\to +\infty$ with $q\ge 2.$ In the case when $q=2$, if $p$ is bigger than $m$, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when $q>2$, we show that existence of global in time solutions always prevails. The case of {\it slowly} decaying density at infinity, i.e. $q\in [0,2)$, is examined in [41].Comment: arXiv admin note: substantial text overlap with arXiv:1911.0604

Uniqueness for fractional parabolic and elliptic equations with drift

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations with a drift.Comment: arXiv admin note: substantial text overlap with arXiv:1306.507

Gradient flow for a class of diffusion equations with Dirichlet boundary data

In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope

Blow-up and global existence for semilinear parabolic equations on infinite graphs

We investigate existence of global in time solutions and blow-up of solutions to the semilinear heat equation posed on infinite graphs. The source term is a general function $f(u)$. We always assume that the infimum of the spectrum of the Laplace operator $λ_1(G)$ on the graph is positive. According to an interaction between the behavior of $f$ close to $0$ and the value $λ_1(G)$, we get the existence of a global in time solution or blow-up of any nonnegative solution, provided that the initial datum is nontrivial.arXiv admin note: text overlap with arXiv:2112.0012

EFFICIENT ELECTROCATALYTIC H2 PRODUCTION BY IMMOBILIZED Co(III)-MYOGLOBIN

The thermodynamics and kinetics of heterogeneous electron transfer (ET) for Co-substituted horse myoglobin (Co-Mb) and its derivatives with ammonia and imidazole as heme axial ligands were studied with cyclic voltammetry on a pyrolytic graphite electrode along with their ability to mediate the electrocatalytic production of H2 . All the proteins experience a non-diffusive electrochemical regime as electrode-bound species. The adsorbed Co-Mb construct was found to carry out the electrocatalytic reduction of water protons to H2 with a good efficiency under anaerobic conditions thus yielding a simple and tunable system for H2 production. Replacement of H2O as Co axial ligand by ammonia and imidazole significantly lowers the catalytic currents for H3O+/H2O reduction to H2. The E°’ values of the Co(III)/Co(II) redox couple for all species are mainly determined by the enthalpic contribution. Differences were found in the kinetics of ET for the different protein adducts due to changes in the activation enthalpies. However, all species share the same distance of about 14 Å from the electrode surface to the Co(III)/Co(II) center determined using the Marcus model, consistent with a non-denaturing adsorption of the protein