Related papers: Partial regularity for type two doubly nonlinear p…
This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…
We prove H\"older regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved.…
In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in…
We establish the local H\"older continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part. The proof involves…
We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations…
We establish partial regularity for vector-valued solutions to parabolic systems where the coefficients are possibly discontinuous with respect to (x,t). More precisely, we assume a VMO-condition with respect to the (x,t) and continuity…
This paper studies H\"older regularity property of bounded weak solutions to a class of strongly coupled degenerate parabolic systems.
In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-\div\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in…
We demonstrate two proofs for the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad p>2,\quad 0<q<p-1. \]…
This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $\Omega…
We establish H\"older estimates for the time derivative of solutions of fully non-linear parabolic equations that does not necessarily have $C^{2,\alpha}$ estimates.
We prove the global existence of small data solution in all space dimension for weakly coupled systems of semi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, nonlinearity terms…
We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert…
We establish local regularity theory for parabolic systems of Uhlenbeck type with $\varphi$-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local H\"older continuity of the gradients,…
We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…
In this paper, we are interested in the regularity of weak solutions $u\colon\Omega_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $\Omega_T$},…
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine…
We prove a local self-improving property for the gradient of very weak solutions to degenerate parabolic double-phase systems. The result is based on a reverse H\"older inequality with constants that are independent of the solution.…
We study a general form of a degenerate or singular parabolic equation $$ u_t-|Du|^{\gamma}\big(\Delta u+(p-2)\Delta_\infty^Nu\big)=0 $$ that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that…
This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…