Related papers: Dissipation vs. quadratic nonlinearity: from a pri…

In this thesis we investigate how the nonlocalities affect the study of different PDEs coming from physics, and we analyze these equations under almost optimal assumptions of the nonlinearity. In particular, we focus on the fractional…

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also…

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose…

Under suitable assumptions on $\beta:\mathbb{R}\!\to\!\mathbb{R}, \,D:\mathbb{R}^d\!\to\!\mathbb{R}^d$ and $b:\mathbb{R}^d\!\to\!\mathbb{R}$, the nonlinear Fokker-Planck equation $u_t-\Delta\beta(u)+{\rm div}(Db(u)u)=0$, in…

We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of…

Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…

Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…

A long-time behavior of solutions to a nonlinear plate model subject to non-conservative and non-dissipative effects and nonlinear damping is considered. The model under study is a prototype for a suspension bridge under the effects of…

In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary…

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in…

We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + \sigma(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\sigma:\mathbb{R}…

We consider nonlinear parabolic equations of the type $$ u_t - div a(x, t, Du)= f(x,t) on \Omega_T = \Omega\times (-T,0), $$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates…

We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-\Delta)^{-1/2}u).$ For definiteness, the problem is posed…

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also…

In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in…

Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of motion, strain, and fractional order…

We investigate the existence of strong solutions to a general class of doubly multivalued and nonlinear evolution equations of second order. The multivalued operators are generated by the subdifferential of nonsmooth potentials that live in…

This paper deals with the following Petrovsky equation with damping and nonlinear source \[u_{tt}+\Delta^2 u-M(\|\nabla u\|_2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u\] under initial-boundary value conditions, where $M(s)=a+…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

In honor of the great Russian mathematician A. N. Kolmogorov, we would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time…