相关论文: Multi-index Based Solution Theory to the $\Phi^4$ …

In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…

We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single…

In this paper, we develop a comprehensive mathematical toolbox for the construction and spectral stability analysis of stationary multiple front and pulse solutions to general semilinear evolution problems on the real line with spatially…

Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this…

We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The…

Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the seventh paper, the usual structural analysis of beams on an elastic foundation…

We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…

We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic…

In [19-20], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic…

We consider the energy critical nonlinear Schr\"odinger equation on periodic domains of the form R^m x T^{4-m} with m=0,1,2,3. Assuming that a certain L^4 Strichartz estimate holds for solutions to the corresponding linear Schr\"odinger…

We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on…

We design the first learned index that solves the dictionary problem with time and space complexity provably better than classic data structures for hierarchical memories, such as B-trees, and modern learned indexes. We call our solution…

In order to prove numerically the global existence and uniqueness of smooth solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori upper bounds on the supremum of the numerical range of the linearized operator. These…

This paper establishes the existence, uniqueness, and global $C^{1,\beta}$ regularity of positive classical solutions to a class of quasilinear Hamilton--Jacobi--Bellman (HJB) equations with Dirichlet boundary conditions on bounded convex…

This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$,…

We prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed point index. Some of the…

In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical…

This paper is devoted to the analysis of periodic solutions of nonlinear control-affine systems with bang-bang controls. Such problems naturally arise in periodic optimal control with constrained inputs, which have, in particular, important…

This is the first part of a two-paper series studying nonlinear Schr\"odinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schr\"odinger equation. Under the assumption that the Fourier…

This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of existence is based on a…