Fujita Exponent Research Articles

Weakly coupled systems of semilinear σ$$ \sigma $$‐evolution equations with friction and visco‐elastic type damping

In the present paper, we are interested to study global (in time) existence of small data Sobolev solutions to the Cauchy problem for a weakly coupled system of two semilinear ‐evolution equations with friction and visco‐elastic type damping, where for . We study model (1.1) (see below) in several cases with respect to the regularity for the data: First, we assume data . By using estimates of Sobolev solutions to related linear models with vanishing right‐hand side, we explain the admissible range of exponents in (1.1) which allow to prove the global (in time) existence of small data Sobolev solutions. Then we suppose that the data belong to , where now , . So the data have additional regularity to the first assumed integrability. We restrict ourselves to data from energy space (on the basis of ) and from energy space with additional suitable higher regularity. We compare in our statements the admissible set of exponents and in the power nonlinearities with the so‐called modified Fujita exponent. Finally, some blow‐up results complete the paper.

The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg group

The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg group

Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential

In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem (wt-Δw)s=λ|x|2sw+wp+f,inRN×(0,+∞),w(x,t)=0,inRN×(-∞,0],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} (w_t-\\Delta w)^s=\\frac{\\lambda }{|x|^{2s}} w+w^p +f, &{}\\quad \ ext {in }\\mathbb {R}^N\ imes (0,+\\infty ),\\\\ w(x,t)=0, &{}\\quad \ ext {in }\\mathbb {R}^N\ imes (-\\infty ,0], \\end{array}\\right. } \\end{aligned}$$\\end{document}where N>2s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N> 2s$$\\end{document}, 0<s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s<1$$\\end{document} and 0<λ<ΛN,s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\lambda <\\Lambda _{N,s}$$\\end{document}, the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{+}(\\lambda , s)$$\\end{document} and of a Fujita-type exponent F(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(\\lambda ,s)$$\\end{document} such that the following holds:Let p>p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>p_+(\\lambda ,s)$$\\end{document}. Then there are not any non-negative supersolutions.Let p<p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<p_+(\\lambda ,s)$$\\end{document}. Then there exist local solutions, while concerning global solutions we need to distinguish two cases:Let 1<p≤F(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 1< p\\le F(\\lambda ,s)$$\\end{document}. Here we show that a weighted norm of any positive solution blows up in finite time.Let F(λ,s)<p<p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(\\lambda ,s)<p<p_+(\\lambda ,s)$$\\end{document}. Here we prove the existence of global solutions under suitable hypotheses.

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations

Abstract In this paper, we are interested in studying the Cauchy problem for a weakly coupled system of two semi-linear structurally damped σ k {\sigma_{k}} -evolution equations, where σ k ≥ 1 {\sigma_{k}\geq 1} for k = 1 , 2 {k=1,2} . Our first purpose involves the proof of global (in time) existence of small data energy solutions by mixing additional L m k {L^{m_{k}}} regularity for the data, where m k ∈ [ 1 , 2 ) {m_{k}\in[1,2)} . We want to point out that in some cases of choosing suitable parameters m k {m_{k}} , with k = 1 , 2 {k=1,2} , the obtained lower bound of one exponent p or q related to power nonlinearities on the right-hand sides is really smaller than the critical exponent, the so-called modified Fujita exponent. The second aim of this paper is to discuss a blow-up result for Sobolev solutions with any different fractional values of σ k ≥ 1 {\sigma_{k}\geq 1} when m 1 = m 2 {m_{1}=m_{2}} .

Blow-up of positive solutions for the semilinear heat equation with a potential

We study nonnegative solutions of the equation in , t>0, under the assumption that satisfies . We establish a new blow-up criterion that depends only on and p. In addition, as the supreme of decreases, we find an interesting phenomenon that the Fujita exponent goes to ∞, in the sense that every nonnegative solution blows up in finite time whenever p>1. Furthermore, we obtain the blow-up rate estimate in the subcritical case. In the end, under the assumption of , we give the refined blow-up estimate of the blow-up solutions near the blow-up time.

On the Fujita Exponent for a Hardy–Hénon Equation with a Spatial–Temporal Forcing Term

The purpose of this work is to analyze the well-posedness and the blow-up of solutions of the higher-order parabolic semilinear equation: $$\begin{aligned} u_t+(-\varDelta )^{d}u=|x|^{\alpha }|u|^{p}+\zeta (t){\textbf{w}}(x) \ \quad \text{ for } (x,t)\in \mathbb {R}^{N}\times (0,\infty ), \end{aligned}$$ where $$d\in (0,1)\cup \mathbb {N}$$ , $$p>1$$ , $$-\alpha \in (0,\min (2d,N))$$ or $$\alpha \geqslant 0$$ and $$\zeta $$ as well as $${\textbf{w}}$$ are suitable given functions. The novelty here regard to previous works lies in considering a forcing term $$\zeta (t){\textbf{w}}(x)$$ depending on both time and space variables, a fractional or high-order Laplace operator $$(-\varDelta )^{d}$$ , and a weighted (possibly singular) nonlinearity $$|x|^{\alpha }|u|^{p}$$ . We prove small Lebesgue data global existence for p larger than a critical value depending on the behavior of $$\zeta $$ . Furthermore, the global existence fails to hold for p less than the critical value under the additional condition $$\int _{\mathbb {R}^N}{} {\textbf {w}}(x)\,\textrm{d}x>0$$ . As a consequence, we derive the explicit value of the Fujita exponent $$p_\textrm{F}(\sigma )$$ in the case $$\zeta (t)=t^\sigma ,\, \sigma >-1$$ .

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.

Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term

Abstract This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

We consider the semilinear diffusion equation ∂ t u = Au + |u| α u in the half-space R N + := R N −1 × (0, +∞), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ′ , 0) = 0 for x ′ ∈ R N −1. We prove that if the symbol of the operator A is of order a|ξ| β near the origin ξ = 0, for some β ∈ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 β/(N + 1). Notice that in the case of the half-space, the exponent β/(N + 1) is smaller than the so-called Fujita exponent β/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).

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.

Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group

We first consider a nonlinear time fractional wave equation with a fractional damping posed on the Heisenberg group with Caputo fractional derivatives that interpolate the heat equation and the wave equation with the linear damping. It extends the Caputo–Wisner, the modified Szabo and the fractional Zener models. The non‐linearity accounts for a nonlinear medium. We present the Fujita exponent for blow‐up which sheds light on the admissible nonlinearity in practice. Then, we establish sufficient conditions ensuring non‐existence of local solutions. using the nonlinear capacity method. Furthermore, we extend the analysisto the case of the system to show the flexibility of the method used.

Fujita exponents for an inhomogeneous parabolic equation with variable coefficients

This paper deals with a Cauchy problem of the inhomogeneous parabolic equation in ), where constants , and . The Japanese brackets ; and the initial data are continuous functions in . We determine the Fujita exponent for global and non‐global solutions of the problem, depending strictly on and , which complete the ones for the nonnegative solutions in J. Math. Anal. Appl. 251 (2000) 624–648 for . It is so interesting that the inhomogeneous term leads to the discontinuity of this critical exponent with respect to at zero.

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.

Fujita exponents for a space–time weighted parabolic problem in bounded domains

This paper deals with a semilinear weighted parabolic problem in general bounded domains, subject to zero Dirichlet boundary conditions, where the weighted functions depend not only on space variable but also on time variable. Fujita exponents for blow-up and global existence of solutions are determined by using semigroup methods and the comparison principle, which are composed by the dimensions of the space domains and the eight exponents in nonlinear coupled sources and the space–time weighted functions.

Fujita exponent and nonexistence result for the Rockland heat equation

This article presents nonexistence results for semilinear parabolic equation with hypoelliptic operator. In particular, we show Fujita exponent for the Rockland heat equation on graded Lie group, which depends on the homogeneous dimension of group and the order of the Rockland operator.

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 heatlike lifespan of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime

Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann-Lemaître-Robertson-Walker spacetimes. We treat the so-called heatlike case where the critical exponent is affected by the Fujita exponent. We show upper bounds of the lifespan of blow-up solutions by distinguishing subcritical and critical cases.