The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “Porous medium equation and fast diffusion equation as gradient systems”

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

Similarity:

We study nonlinear diffusion problems of the form u t = u x x + f ( u ) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f ( u ) of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any f ( u ) which is C 1 and satisfies f ( 0 ) = 0 , we show that the omega limit set ω ( u ) of every bounded positive solution is determined by a stationary...

Blow up for a completely coupled Fujita type reaction-diffusion system

Noureddine Igbida, Mokhtar Kirane (2002)

Colloquium Mathematicae

Similarity:

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form u - Δ ( a 11 u ) = h ( t , x ) | v | p , v - Δ ( a 21 u ) - Δ ( a 22 v ) = k ( t , x ) | w | q , w - Δ ( a 31 u ) - Δ ( a 32 v ) - Δ ( a 33 w ) = l ( t , x ) | u | r , for x N , t > 0, p > 0, q > 0, r > 0, a i j = a i j ( t , x , u , v ) , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for x N , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...

Lyapunov functions and L p -estimates for a class of reaction-diffusion systems

Dirk Horstmann (2001)

Colloquium Mathematicae

Similarity:

We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, ε c = k c Δ c - f ( c ) c + g ( a , c ) , x ∈ Ω, t > 0, for Ω N , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform L p -estimates.

Nonlinear diffusion equations with perturbation terms on unbounded domains

Kurima, Shunsuke

Similarity:

This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term u t + ( - Δ + 1 ) β ( u ) + G ( u ) = g in Ω × ( 0 , T ) in an unbounded domain Ω N with smooth bounded boundary, where N , T > 0 , β , is a single-valued maximal monotone function on , e.g., β ( r ) = | r | q - 1 r ( q > 0 , q 1 ) and G is a function on which can be regarded as a Lipschitz continuous operator from ( H 1 ( Ω ) ) * to ( H 1 ( Ω ) ) * . The present work establishes existence and estimates for the above problem.

Existence and upper semicontinuity of uniform attractors in H ¹ ( N ) for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

Similarity:

We prove the existence of uniform attractors ε in the space H ¹ ( N ) for the nonautonomous nonclassical diffusion equation u t - ε Δ u t - Δ u + f ( x , u ) + λ u = g ( x , t ) , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ε ε [ 0 , 1 ] at ε = 0 is also studied.

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

Similarity:

In this paper we examine self-similar solutions to the system u i t - d i Δ u i = k = 1 m u k p k i , i = 1,…,m, x N , t > 0, u i ( 0 , x ) = u 0 i ( x ) , i = 1,…,m, x N , where m > 1 and p k i > 0 , to describe asymptotics near the blow up point.

Vectorial quasilinear diffusion equation with dynamic boundary condition

Nakayashiki, Ryota

Similarity:

In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S) ε with a nonnegative constant ε , and for any ε 0 , (S) ε can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain Ω , and the parabolic equation on the boundary Γ : = Ω , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Honghui Yin, Zuodong Yang (2012)

Annales Polonici Mathematici

Similarity:

Our main purpose is to establish the existence of a positive solution of the system ⎧ - p ( x ) u = F ( x , u , v ) , x ∈ Ω, ⎨ - q ( x ) v = H ( x , u , v ) , x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where Ω N is a bounded domain with C² boundary, F ( x , u , v ) = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] , H ( x , u , v ) = λ q ( x ) [ g ( x ) b ( v ) + h ( u ) ] , λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and - p ( x ) u = - d i v ( | u | p ( x ) - 2 u ) is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

Homogenization of a three-phase composites of double-porosity type

Ahmed Boughammoura, Yousra Braham (2021)

Czechoslovak Mathematical Journal

Similarity:

In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size ε β ( ε > 0 and β > 0 ) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order ε 2 (the so-called double-porosity type scaling) while the matrix material has a conductivity...

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process ( K ( t ) , i ( t ) , Y ( t ) ) on ( 𝕋 2 × { 1 , 2 } × 2 ) , where 𝕋 2 is the two-dimensional torus. Here ( K ( t ) , i ( t ) ) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y ( t ) is an additive functional of K , defined as 0 t v ( K ( s ) ) d s , where | v | 1 for small k . We prove that the rescaled process ( N ln N ) - 1 / 2 Y ( N t ) converges in distribution to a two-dimensional Brownian motion. As a consequence,...

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

Similarity:

We consider the semilinear Lane–Emden problem where p > 1 and Ω is a smooth bounded domain of 2 . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ( p ) , as p + . Among other results we show, under some symmetry assumptions on Ω , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p + , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

Quasi-diffusion solution of a stochastic differential equation

Agnieszka Plucińska, Wojciech Szymański (2007)

Applicationes Mathematicae

Similarity:

We consider the stochastic differential equation X t = X + 0 t ( A s + B s X s ) d s + 0 t C s d Y s , where A t , B t , C t are nonrandom continuous functions of t, X₀ is an initial random variable, Y = ( Y t , t 0 ) is a Gaussian process and X₀, Y are independent. We give the form of the solution ( X t ) to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that ( X t ) is a quasi-diffusion proces.

On the r -free values of the polynomial x 2 + y 2 + z 2 + k

Gongrui Chen, Wenxiao Wang (2023)

Czechoslovak Mathematical Journal

Similarity:

Let k be a fixed integer. We study the asymptotic formula of R ( H , r , k ) , which is the number of positive integer solutions 1 x , y , z H such that the polynomial x 2 + y 2 + z 2 + k is r -free. We obtained the asymptotic formula of R ( H , r , k ) for all r 2 . Our result is new even in the case r = 2 . We proved that R ( H , 2 , k ) = c k H 3 + O ( H 9 / 4 + ε ) , where c k > 0 is a constant depending on k . This improves upon the error term O ( H 7 / 3 + ε ) obtained by G.-L. Zhou, Y. Ding (2022).

Maximal upper asymptotic density of sets of integers with missing differences from a given set

Ram Krishna Pandey (2015)

Mathematica Bohemica

Similarity:

Let M be a given nonempty set of positive integers and S any set of nonnegative integers. Let δ ¯ ( S ) denote the upper asymptotic density of S . We consider the problem of finding μ ( M ) : = sup S δ ¯ ( S ) , where the supremum is taken over all sets S satisfying that for each a , b S , a - b M . In this paper we discuss the values and bounds of μ ( M ) where M = { a , b , a + n b } for all even integers and for all sufficiently large odd integers n with a < b and gcd ( a , b ) = 1 .

Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling, Masahiko Shimojō (2019)

Mathematica Bohemica

Similarity:

We consider solutions of quasilinear equations u t = Δ u m + u p in N with the initial data u 0 satisfying 0 < u 0 < M and lim | x | u 0 ( x ) = M for some constant M > 0 . It is known that if 0 < m < p with p > 1 , the blow-up set is empty. We find solutions u that blow up throughout N when m > p > 1 .