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We consider an inverse boundary value problem for the hyperbolic partial differential equation $ (-i\partial_{t} + A_{0}(t,x))^2 u(t,x) - \sum_{j=1}^n (-i\partial_{x_j} + A_{j}(t,x))^2 u(t,x) + V(t,x)u(t,x) = 0 $ with time dependent vector…
In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+\chi(t)|Du|^\Gamma-f(u)u_t=$ in $\Omega\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions,…
In this article we study the quasi-linear equation \[\mathrm{div}\, \mathcal A(x,u,\nabla u)=\mathcal B(x,u,\nabla u)\quad \text{in }\Omega,\qquad u\in H^{1,p}_{loc}(\Omega;w_1dx)\] where $\mathcal A$ and $\mathcal B$ are functions…
In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…
For nonautonomous, nonuniformly elliptic integrals with so-called $(p,q)$-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for H\"older continuous minimizers under improved…
We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…
This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary,…
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is --div((1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u) -- div(c(x)|u| p--2 u) = f in $\Omega$, (1 + |$\nabla$u| 2)…
We prove that bounded weak solutions to degenerate parabolic double-phase equations of $p$-Laplace type are locally H\"older continuous. The proof is based on phase analysis and methods for the $p$-Laplace equation. In particular, the phase…
In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a…
We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-\Delta_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The…
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 establish the local Lipschitz regularity in space for the viscosity solutions to the parabolic double phase equation of the form \[ \smash{\partial_{t}u-\operatorname{div} \left(|Du|^{p-2}D u+a(z)|D u|^{q-2}D u\right)=f(z, Du)} \] by…
In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…
Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods…
This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-\Delta)^{s_1}_{p}u+\ba(-\Delta)^{s_2}_{q}u = \la a(x)|u|^{\delta-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on }…
This paper investigates the existence of weak solutions of biquasilinear boundary value problem for a coupled elliptic-parabolic system of divergence form with discontinuous leading coefficients. The mathematical framework addressed in the…
We study local and global higher integrability properties for quasiminimizers of a class of double-phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure…
This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…
In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian \begin{equation*} \label{1} \begin{cases} (-\Delta)^{s}u = \lambda u^{-q} + u^{2^*_s-1}, \quad…