Related papers: Large field problem in coercive singular PDEs

We consider the approximation of singularly perturbed linear second-order boundary value problems by $hp$-finite element methods. In particular, we include the case where the associated differential operator may not be coercive. Within this…

Consider the following time-dependent stable-like operator with drift $$ \mathscr{L}_t\varphi(x)=\int_{\mathbb{R}^d}\big[\varphi(x+z)-\varphi(x)-z^{(\alpha)}\cdot\nabla\varphi(x)\big]\sigma(t,x,z)\nu_\alpha(d z)+b(t,x)\cdot\nabla…

We prove sharp $L^p$ estimates for a singular transport equation by building what we call a \emph{cascading solution}; the equation studies the combined effect of multiplying by a bounded function and application of the Hilbert transform.…

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric,…

We consider a strongly nonlinear differential equation of the following general type $$(\Phi(a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text{a.e. on $[0,T]$}$$ where $f$ is a Carath\'edory function, $\Phi$ is a strictly increasing…

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\…

In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean…

For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…

In this article we study local and global properties of positive solutions of $-\Delta_mu=|u|^{p-1}u+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, with $m>1$, $p,q>0$ and $M\in\mathbb R$. Following some ideas used in…

Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2,…

In this paper, we establish a priori estimates for the positive solutions to a higher-order fractional Laplace equation on a bounded domain by a blowing-up and rescaling argument. To overcome the technical difficulty due to the high-order…

In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $\Omega \subset \mathbb{R}^N$, we show that a solution $\phi \in…

This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-\Delta u=u^p+M|\nabla u|^q$ in $\Omega\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the…

Discrete approximations to the equation \begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A'(x)+H(x)) u^{(1)} + B(x) u = f, \; x\in[0,1] \end{equation*} are considered. This is an extension of the Sturm-Liouville case…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

For a classical elliptic pseudodifferential operator P of order m>0 on a closed manifold X, such that the eigenvalues of the principal symbol p_m(x,\xi) have arguments in \,]\theta,\phi [\, and \,]\phi, \theta +2\pi [\, (\theta <\phi…

We examine quintessence models for dark energy in which the scalar field, $\phi$, evolves near the vicinity of a local maximum or minimum in the potential $V(\phi)$, so that $V(\phi)$ be approximated by a quadratic function of $\phi$ with…

In this paper we derive some a priori estimates for a class of linear coagulation equations with particle fluxes towards large size particles. The derived estimates allow us to prove local well posedness for the considered equations. Some…

We investigate the realization of a myriad of general local and nonlocal inhomogeneous elliptic and parabolic boundary value problems over classes of irregular regions. We present a unified approach in which either local or nonlocal…

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm…