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In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers…
In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under…
By introducing a new classification of the growth rate of exponential functions, singular solutions for semilinear elliptic equations in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model…
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 global existence and regularity of H\"older continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
We study regularity issues and the limiting behavior as $p\to\infty$ of nonnegative solutions for elliptic equations of $p-$Laplacian type ($2 \leq p< \infty$) with a strong absorption: $$ -\Delta_p u(x) + \lambda_0(x) u_{+}^q(x) = 0 \quad…
In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
We consider a process given as the solution of a one-dimensional stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. H\"older continuity of the Lebesgue density of…
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione…
In this paper we study the discrete coagulation--fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong.…
Local H\"older regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling…
Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research…
In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest…
We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla…
In this article, we examine the H\"older regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator $\fp + \fqs$ with singular data, under the minimal assumption that $p> sq$. The regularity…
In this paper, we prove local H\"older continuity for the spatial gradient of weak solutions to $$u_t - \text{div} (|\nabla u|^{p-2}\nabla u) + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+ps}} \ dy…
The article addresses the convergence of implicit and semi-implicit, fully discrete approximations of a class of nonlinear parabolic evolution problems. Such schemes are popular in the numerical solution of evolutions defined with the…
This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
In this paper we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.