相关论文: Existence of similarity profiles for diffusion equ…
We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as…
In this article we propose a generalization of the theory of diffusion approximation for random ODE to a nonlinear system of random Schr\"{o}dinger equations. This system arises in the study of pulse propagation in randomly birefringent…
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady…
We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the…
This study handles spatial three-dimensional solution of the nonlinear diffusion equation without particular initial conditions. The functional behavior of the equation and the concentration have been studied in new ways. An auxiliary…
A constructive approach to theory of diffusion processes is proposed, which is based on application of both the symmetry analysis and method of modelling functions. An algorithm for construction of the modelling functions is suggested. This…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…
A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be…
We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ in the range of exponents…
We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of…
Boundary characteristic point regularity is studied for a class of semilinear heat equations and an ODE criterion of regularity is obtained. Extensions to higher-order semilinear parabolic problems are discussed.
The goal of this work is to establish the global existence of nonnegative classical solutions in all dimensions for a system of highly nonlinear reaction-diffusion equations. We address the case for different diffusion coefficients and the…
A system of equations of the reaction-diffusion type is studied in the framework of both the direct and the inverse prolongation structure. We find that this system allows an incomplete prolongation Lie algebra, which is used to find the…
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on…
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
Diffusion processes with boundaries are models of transport phenomena with wide applicability across many fields. These processes are described by their probability density functions (PDFs), which often obey Fokker-Planck equations (FPEs).…
Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical…
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is…
We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^{\sigma}u^m, $$ posed for $(x,t)\in\real\times(0,T)$, with $m>1$ and $\sigma>0$. In strong contrast…
We consider solvability of the generalized reaction-diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction-diffusion…