Critical Fujita Exponent Research Articles

Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive solutions and obtain an optimal Fujita-type result: any positive solution blows up in finite time if $p\leq 1+\frac{2}{n}$ or $q\leq 1+\frac{1}{n+1}$, while global classical positive solutions exist for suitably small initial data when $p>1+\frac{2}{n}$ and $q> 1+\frac{1}{n+1}$. Although finite time blow-up cannot be produced by the gradient term alone and should be considered as an effect of the source term $|u|^p$, this result shows that the gradient term induces an interesting phenomenon of discontinuity of the critical Fujita exponent, jumping from $p=1+\frac{2}{n}$ to $p=\infty$ as $q$ reaches the value $1+\frac{1}{n+1}$ from above. Next, we investigate the case of sign-changing solutions and show that if $p\le 1+\frac{2}{n}$ or $0<(q-1)(np-1)\le 1$, then the solution blows up in finite time for any nontrivial initial data with nonnegative mean. Finally, a Fujita-type result, with a different critical exponent, is % also obtained for sign-changing solutions to the inhomogeneous version of this problem.

First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities

We investigate the Cauchy problem for the nonlinear damped wave equation $u_{tt}-\Delta u +u_t = |u|^p+|\nabla u|^q +w(x)$, where $N\geq 1$, $p, q>1$, $w\in L^1_{loc}(\mathbb{R}^N)$, $w\geq 0$ and $w\not\equiv 0$. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term $|\nabla u|^q$ induces a phenomenon of discontinuity of the Fujita critical exponent.

Finite-time blow-up for inhomogeneous parabolic equations with nonlinear memory

In this paper, we study the effect of an inhomogeneity on the finite-time blow-up of solutions to the nonlinear heat equation where , and p>1. It is well known that in the homogeneous case , the Fujita critical exponent is given by In the case and , we prove that the critical exponent is equal to ∞, which means that for all p>1, we have a finite-time blow-up.

Critical exponents for a non‐Newtonian polytropic filtration equation with weighted nonlocal inner sources

This article considers the Cauchy problem for a non‐Newtonian polytropic filtration equation with weighted nonlocal inner sources where N ≥ 1, , 0&lt;m ≤ 1, q&gt;1, r ≥ 1, , and r+s&gt;1. We first obtain a new critical Fujita exponent by virtue of the auxiliary function method and the forward self‐similar solution and then determine the second critical exponent to classify global and nonglobal solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of nonglobal solution are derived.

New critical exponents for a doubly singular parabolic equation

ABSTRACT This paper deals with the Cauchy problem for a doubly singular parabolic equation with nonlocal inner source where , , , q>1, , and r + s>1. We first obtain a new critical Fujita exponent by virtue of the auxiliary function method and the forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.

New critical exponents, large time behavior, and life span for a fast diffusive p-Laplacian equation with nonlocal source

This paper mainly investigate positive solutions of the Cauchy problem for a fast diffusive p-Laplacian equation with nonlocal source $$\begin{aligned} u_{t}=\Delta _pu+\left( \,\,\int \limits _{{\mathbb {R}}^N}u^q(y,t)\mathrm{d}y\right) ^{\frac{r-1}{q}}u^{s+1},\quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{aligned}$$ where $$N\ge 1$$ , $$\frac{2N}{N+1}<p<2$$ , $$q>1$$ , $$r\ge 1$$ , $$0\le s<\left( 1+\frac{1}{N}\right) p-2$$ and $$r+s>1$$ . We obtain the new critical Fujita exponent by virtue of the auxiliary function method and forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.

Asymptotic behavior of solutions to a class of coupled nonlinear parabolic systems

This paper studies the Cauchy problem to a class of coupled nonlinear parabolic systems and investigates the asymptotic behavior of solutions to the problem. The blow-up theorem of Fujita type is established by the integral estimation and suitable supersolutions. Moreover, the critical Fujita exponent determined by the diffusion and the spatial dimension is given.

Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent

In this paper, we deal with a time-weighted parabolic system coupled via nonlocal sources. Firstly, we give the critical Fujita exponent of solutions. Secondly, we propose the time-weighted blow-up profiles of solutions. Thirdly, the simultaneous and non-simultaneous blow-up of solutions are discussed under some conditions on the initial data. Fourthly, besides blow-up rates, we classify total blow-up and single point blow-up of solutions. At last, upper and lower bounds of blow-up time are studied according to precise blow-up phenomena.

The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions

This paper is concerned with the lifespan estimate of classical solutions with small initial data to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent. We establish the following sharp upper bound of the lifespanT(ε)≤exp⁡(Cε−2n) in higher dimensions (n≥4), by using the heat kernel as the test function. Then, together with the previous results, a complete result on the sharp lower and upper bound estimates is obtained in this case.

Blow-up theorems of Fujita type for a semilinear parabolic equation with a gradient term

This paper deals with the existence and non-existence of the global solutions to the Cauchy problem of a semilinear parabolic equation with a gradient term. The blow-up theorems of Fujita type are established and the critical Fujita exponent is determined by the behavior of the three variable coefficients at infinity associated to the gradient term and the diffusion–reaction terms, respectively, as well as the spacial dimension.

Second critical exponent for a nonlinear nonlocal diffusion equation

This paper studies the Cauchy problem for a nonlocal diffusion equation u t = J ∗ u − u + u p . We determine the second critical exponent, namely the optimal decay order for initial data at space infinity to distinguish global and blow-up solutions in the “super Fujita” range. Similar to the critical Fujita exponent obtained very recently by Alfaro (2017), we find that the second critical exponent also relies heavily on the behavior of the Fourier transform of the kernel function J .

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations ut=Δu+eαtvp and vt=Δv+eβtuq, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Fujita phenomenon in higher-order parabolic equation with nonlocal term

This paper considers the Cauchy problem for a higher-order semilinear parabolic equation with nonlocal term in , where , , , and . We obtain the critical Fujita exponent to the problem, in the sense that every nontrivial solution blows up in finite time whenever , and there are both global and non-global solutions if .

Critical Fujita exponents to a class of non-Newtonian filtration equations with fast diffusion

We consider the Cauchy problem to a class of fast-diffusion non-Newtonian filtration equations. Besides the usual degeneracy in the fast-diffusion non-Newtonian filtration, the equation is degenerate or singular at infinity, depending on the sign of the parameter related to the coefficient of diffusion. Fujita type theorems are established and the critical Fujita exponent is determined. Specially, we also prove that the nontrivial solution blows up in a finite time on the critical situation.

Existence and nonexistence of global solutions for a semilinear reaction–diffusion system

This paper is concerned with the blow-up and global existence of nonnegative solutions to the following Cauchy problemut−Δu=vp,t>0,x∈RN,vt−Δv=a(x)uq,t>0,x∈RN,u(x,0)=u0(x),v(x,0)=v0(x),x∈RN, where the constants p,q>0 and a(x)≩0 is on the order |x|m as |x|→∞, m∈R. The Fujita critical exponent is determined when m≥0, and some results of global existence of solution under some assumptions when m<0 are also obtained. The results extend those in Escobedo and Herrero (1991) [9] and indicate that m affects the Fujita critical exponent.

On the Properties of Solutions of Multidimensional Nonlinear Filtration Problem with Variable Density and Nonlocal Boundary Condition in the Case of Fast Diffusion

The conditions of global existence of solutions of a nonlinear filtration problem in an inhomogeneous medium are investigated in this paper. Various techniques such as the method of standard equations, selfsimilar analysis and the comparison principle are used to obtain results. The influence of inhomogeneous medium on the evolution process is analyzed. The critical global existence exponent and the critical Fujita exponent are obtained. Asymptotic behavior of solutions in the case of the global solvability is established.

Lifespan of solutions to the damped wave equation with a critical nonlinearity

In the present paper, we study a lifespan of solutions to the Cauchy problem for semilinear damped wave equations(DW){∂t2u−Δu+∂tu=f(u),(t,x)∈[0,T(ε))×Rn,u(0,x)=εu0(x),x∈Rn,∂tu(0,x)=εu1(x),x∈Rn, where n≥1, f(u)=±|u|p−1u or |u|p, p≥1, ε>0 is a small parameter, and (u0,u1) is a given initial data. The main purpose of this paper is to prove that if the nonlinear term is f(u)=|u|p and the nonlinear power is the Fujita critical exponent p=pF=1+2n, then the upper estimate to the lifespan is estimated byT(ε)≤exp⁡(Cε−p) for all ε∈(0,1] and suitable data (u0,u1), without any restriction on the spatial dimension. Our proof is based on a test-function method utilized by Zhang [35]. We also prove a sharp lower estimate of the lifespan T(ε) to (DW) in the critical case p=pF.

Critical exponents for the multidimensional polytrophic filtration problem with a variable coefficients and nonlocal boundary condition

In this paper we study the global solvability and no solvability conditions of a multidimensional nonlinear problem of non-Newtonian filtration with nonlocal boundary condition in the slow diffusion case. Establish the critical global existence exponent and critical Fujita exponent of nonlinear filtration problem in inhomogeneous medium, which play an important role in the study of qualitative properties of nonlinear models of reaction - diffusion, thermal conductivity, filtering, and other physical, chemical, and biological processes. In the global solvability case the main terms of the asymptotic of solutions is obtained.

Estimates and Asymptotic of Self-similar Solutions to a Nonlinear Filtration Problem with Variable Density and Nonlocal Boundary Conditions

In this paper we study the global solvability or nosolvability of a nonlinear filtration problem with nonlinear flux boundary condition in the fast diffusion case. The critical global existence and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions are obtained.

Cauchy problem for a class of fourth-order semilinear pseudohyperbolic equations with structural damping

A sufficient condition for the existence of global solutions is established, and the rate of decay of solutions to the Cauchy problem for a semilinear pseudohyperbolic equation with structural damping is found. The nonexistence of global solutions is also investigated. An analogue of the critical Fujita exponent is obtained for the considered problem.