Critical Fujita Exponent Research Articles

Asymptotic behavior of solutions to multi-dimensional semilinear parabolic equations with general diffusion coefficients

In this paper, we study asymptotic behavior of solutions to the Cauchy problem of a class of multi-dimensional semilinear parabolic equations with general diffusion coefficients. It is shown that the critical Fujita exponent is determined by the spatial dimension and the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is proved to belong to the blowing-up case.

A higher order evolution inequality with a gradient term in the exterior of the half-ball

<abstract><p>We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form $ |x|^a $ is allowed in front of the power nonlinearity. When $ a > -2 $, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.</p></abstract>

Blow-up versus global existence of solutions for reaction–diffusion equations on classes of Riemannian manifolds

It is well known from the work of Bandle et al. (J Differ Equ 251:2143–2163, 2011) that the Fujita phenomenon for reaction–diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space {mathbb {H}}^N, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom Lambda of the L^2 spectrum of -Delta is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by Lambda . Such nonlinearities are time-independent, in contrast to the ones studied in Bandle et al. (2011). As a consequence of our results we show that, on a class of manifolds much larger than the case M={mathbb {H}}^N considered in Bandle et al. (2011), solutions to (1.1) with power nonlinearity f(u)=u^p, p>1, and corresponding to sufficiently small data, are global in time. Though qualitative similarities with similar problems in bounded, Euclidean domains can be seen in the results, the methods are significantly different because of noncompact setting dealt with.

First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities

We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation iut+Δu=λ|u|p+μ|∇u|q+w(x),t>0,x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} iu_t+\\Delta u=\\lambda |u|^p+\\mu |\ abla u|^q+w(x),\\quad t>0,\\, x\\in {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 1, p,q>1, lambda ,mu in {mathbb {C}}, lambda ne 0, and u(0,cdot ), win L^1_{mathrm{loc}}({mathbb {R}}^N,{mathbb {C}}). We consider both the cases where mu =0 and mu ne 0, respectively. We establish existence/nonexistence of global weak solutions. In each studied case, we compute the critical exponents in the sense of Fujita, and Lee and Ni. When mu ne 0, we show that the nonlinearity |nabla u|^q induces an interesting phenomenon of discontinuity of the Fujita critical exponent.

Fujita critical exponent for hyperbolic‐type inequalities with mixed nonlinearities on the half space

We consider the nonlinear hyperbolic‐type inequality under the Dirichlet‐type boundary condition where , and . An optimal Fujita‐type result is obtained for this problem. Namely, we prove that when belongs to a certain functional space , and or , then there exists no global weak solution, while global solutions exist for some when and . The obtained result shows an interesting phenomenon of discontinuity of the Fujita critical exponent, jumping from to as reaches the value from above.

On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\mathbb R^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\mathbb R^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\mathbb R^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\mathbb R^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.

The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory

<abstract><p>In this article, we focus on the blow-up problem of solution to Caputo-Hadamard fractional diffusion equation with fractional Laplacian and nonlinear memory. By virtue of the fundamental solutions of the corresponding linear and nonhomogeneous equation, we introduce a mild solution of the given equation and prove the existence and uniqueness of local solution. Next, the concept of a weak solution is presented by the test function and the mild solution is demonstrated to be a weak solution. Finally, based on the contraction mapping principle, the finite time blow-up and global solution for the considered equation are shown and the Fujita critical exponent is determined. The finite time blow-up of solution is also confirmed by the results of numerical experiment.</p></abstract>

Asymptotic behavior of solutions to porous medium equations with boundary degeneracy

This article concerns the asymptotic behavior of solutions to a class of one-dimensional porous medium equations with boundary degeneracy on bounded and unbounded intervals. It is proved that the degree of degeneracy, the exponents of the nonlinear diffusion, and the nonlinear source affect the asymptotic behavior of solutions. It is shown that on a bounded interval, the problem admits both nontrivial global and blowing-up solutions if the degeneracy is not strong; while any nontrivial solution must blow up if the degeneracy is strong enough. For the problem on an unbounded interval, the blowing-up theorems of Fujita type are established. The critical Fujita exponent is finite if the degeneracy is not strong, while infinite if the degeneracy is strong enough. Furthermore, the critical case is proved to be the blowing-up case if it is finite. For more information see https://ejde.math.txstate.edu/Volumes/2021/96/abstr.html

Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M:{ut−LMu=f(u),x∈M,t>0,u(0,x)=u0(x),x∈M, for u0≥0, where LM is a sub-Laplacian of M. In the case when M is a connected unimodular Lie group G, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with u0≢0, blow up in finite time if and only if 1<p≤pF:=1+2/D when f(u)≃up, where D is the global dimension of G. In the case 1<p<pF and when f:[0,∞)→[0,∞) is a locally integrable function such that f(u)≥K2up for some K2>0, we also show that the differential inequalityut−LMu≥f(u) does not admit any nontrivial distributional (a function u∈Llocp(Q) which satisfies the differential inequality in D′(Q)) solution u≥0 in Q:=(0,∞)×G. Furthermore, in the case when G has exponential volume growth and f:[0,∞)→[0,∞) is a continuous increasing function such that f(u)≤K1up for some K1>0, we prove that the Cauchy problem has a global, classical solution for 1<p<∞ and some positive u0∈Lq(G) with 1≤q<∞. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M.

A general blow-up result for a degenerate hyperbolic inequality in an exterior domain

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

Asymptotic results and critical Fujita exponent in parabolic equations with nonlocal nonlinearity

This paper deals with the parabolic equations coupled via nonlocal multiple nonlinearity. We determine the critical Fujita exponents for blow-up solutions, which are influenced by the time coefficients. Moreover, we obtain the complete and optimal classification on simultaneous and non-simultaneous blow-up of solutions. Then simultaneous blow-up rates and estimates on blow-up time are studied according to the precise blow-up phenomena for all of the dimensions of the space domain.

Fujita type theorem for a class of coupled quasilinear convection\u2013diffusion equations

In this paper, we establish the Fujita type theorem for a homogeneous Neumann outer problem of the coupled quasilinear convection–diffusion equations and formulate the critical Fujita exponent. Besides, the influence of diffusion term, reaction term, and convection term on the global existence and the blow-up property of the problem is revealed. Finally, we discuss the large time behavior of the solution to the outer problem in the critical case and describe the asymptotic behavior of the solution.

On the critical behavior for inhomogeneous parabolic differential inequalities in the half-space

We consider the differential inequality with a nonlinear memory ∂tu−Δu≥1Γ(σ)∫0t(t−s)σ−1|u(s,x)|pds+w(x),t>0,x∈R+N, where N≥2, p>1, σ>0, and w⁄≡0. Under certain conditions on the initial value and the inhomogeneous term w, we show that the Fujita critical exponent is given by pσ∗=∞, for all σ>0. Next, we consider the limit case of the above problem as σ→0+, i.e. ∂tu−Δu≥|u|p+w(x),t>0,x∈R+N. A discontinuity phenomenon of the critical exponent is observed. Namely, we show that the Fujita critical exponent for the limit problem is equal to N+1N−1, which is different to limσ→0+pσ∗.

A general nonexistence result for inhomogeneous semilinear wave equations with double damping and potential terms

Abstract We investigate the large-time behavior of solutions for a class of inhomogeneous semilinear wave equations involving double damping and potential terms. Namely, we first establish a general criterium for the absence of global weak solutions. Next, some special cases of potential and inhomogeneous terms are studied. In particular, when the inhomogeneous term depends only on the variable space, the Fujita critical exponent and the second critical exponent in the sense of Lee and Ni are derived.

Existence and Nonexistence of the Solutions to the Cauchy Problem of Quasilinear Parabolic Equation with a Gradient Term

This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.

Critical Fujita Exponent for the Porous Medium Equation in ${\mathbb R}^{{N}}$ with hole

We study the large time behavior of solutions to the Neumann Problem of the porous medium equation. It is shown that the critical Fujita exponent is determined not only by the spatial dimension and the nonlinear exponent, but also by the coefficient $k$ of the first-order term.

Fujita exponent for an inhomogeneous pseudoparabolic equation

We consider the Cauchy problem of the inhomogeneous pseudoparabolic equation ut−kΔut=Δu+|u|p+ω(x) in ℝn with nonnegative initial value u0(x), where k>0 and p>1. When the equation is homogeneous, i.e., ω≡0, it was shown by Cao, Yin and Wang (2009) that the critical Fujita exponent pc is given by pc=1+2∕n, which means for p∈(1,pc), the distribution of the initial data has no effect on the blow-up phenomena; for p∈(pc,∞), the distribution of the initial data does have effect on the blow-up phenomena. We study the effect of the inhomogeneous term ω(x) on the critical Fujita exponent pc, and we show pc=∞ if n=1,2, and pc=1+2∕(n−2) if n≥3.

Global existence of solutions for a p‐Laplacian equation with nonlocal Fisher–KPP type reaction terms

This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p‐Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent.

Fast diffusion in a porous building material with a nonlocal source

Abstract Fast diffusion is of great importance in concrete solidification, and this paper suggests a modified Fick law to deduce a fast diffusion equation with a nonlocal source. Its global and non-global solutions are obtained, and the critical Fujita exponent and second critical exponent are discussed. A numerical example is given to elucidate the basic properties of the fast diffusion process. The theoretical results shed a promising light on an extremely fast solidification of a concrete in building an artificial island or an urgent repairing in an accident.

Uniform Boundedness and Global Existence of Solutions to a Quasilinear Diffusion Equation with Nonlocal Fisher-KPP Type Reaction Term

This paper deals with the Cauchy problem and Neumann initial boundary value problem for a quasilinear diffusion equation with nonlocal Fisher-KPP type reaction terms. We establish the uniform boundedness and global existence of solutions to the problems by using multipliers technique and modified Moser's iteration argument for some ranges of parameters. Moreover, the ranges of parameters have similar structure to that of the classical critical Fujita exponent.