Fujita Exponent Research Articles

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.

Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

This paper deals with Cauchy problems of pseudo-parabolic equations with inhomogeneous terms. The aim of the paper is to study the influence of the inhomogeneous term on the asymptotic behavior of solutions. We at first determine the critical Fujita exponent and then give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Furthermore, the precise estimate of life span for the blow-up solution is obtained. Our results show that the asymptotic behavior of solutions is seriously affected by the inhomogeneous term.

Fujita-type phenomenon of nonlinear coupled nonlocal diffusion system

This paper studies a Cauchy problem for a nonlinear coupled nonlocal diffusion system ut=J⁎u−u+vp, vt=J⁎v−v+uq. We first deal with the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. It turns out that the results are coincident with the classical heat system. We not only generalize the results in the scalar case obtained by J. García-Melián and F. Quirós (2010) [6] to the case of a system, but also delete the monotonicity condition on the kernel function J. The key techniques are modifying the rescaled test function method with a new test function rather than the eigenfunction of the nonlocal operator to deal with blow-up of the solution, and selecting a suitable Banach space via the estimates obtained by J. Terra and N. Wolanski (2011) [13] to obtain the global solutions.

Second critical exponent for a higher-order semilinear parabolic system

This paper studies a higher-order semilinear parabolic system. We obtain the second critical exponent to characterize the critical space-decay rate of the initial data in the co-existence parameter region of global and non-global solutions. Together with the critical Fujita exponent established by Pang et al. (2006), this gives a clear and complete picture to the Fujita phenomena in the coupled higher-order semilinear parabolic system.

The critical Fujita exponent for a diffusion equation with a potential term

We study the initial-boundary-value problem of the diffusion equation u t = Δu m − V (x)u m + u p in a conelike domain D = [1,∞) × Ω, where V (x) ~ ω 2 |x| −2 with ω 2 > 0. Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω, and let l denote the positive root of l 2 + (n − 2)l = ω 1 + ω 2. We prove that if m < p ≤ m + 2/(n + l), then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if p > m + 2/(n + l), then the problem has global solutions for some $ {u_0}\gneq 0 $ .

Cauchy problem for the non-Newtonian polytropic filtration equation with a localized reaction

This paper is concerned with the Cauchy problem for the non-Newtonian polytropic filtration equation with a localized reactionwhere , , and is a compactly supported function. By constructing various self-similar super- and subsolutions, we give the critical global existence exponent and critical Fujita exponent . Moreover, under some suitable conditions, we obtain the results of blow-up rate and blow-up set for the non-global solution.

Blow-up in the Parablic Problems under Nonlinear Boundary Conditions

The paper deals with a degenerate and singular parabolic equation with nonlinear boundary condition. We first get the behavior of the solution at infinity, and establish the critical global existence exponent and critical Fujita exponent for the fast diffusive equation, furthermore give the blow-up set and upper bound of the blow-up rate for the nonglobal solutions

Fujita Exponent for a Nonlinear Degenerate Parabolic Equation with Localized Source

This paper is devoted to understand the blow-up properties of reaction-diffusion equations which combine a localized reaction term with nonlinear diffusion. In particular, we study the critical exponent of ap-Laplacian equation with a localized reaction. We obtain the Fujita exponentqcof the equation.

A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.

Fujita phenomena in nonlinear pseudo-parabolic system

This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut − Δu −αΔut = vp, vt−Δv−αΔvt = uq with p, q ⩾ 1 and pq > 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the Lp-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.

Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy

This paper concerns the asymptotic behavior of solutions to a semilinear parabolic equation with boundary degeneracy. It is proved that for the problem in a bounded domain with a homogeneous boundary condition, there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong, while the nontrivial solution must blow up in a finite time if the degeneracy is strong enough. For the problem in an unbounded domain, blowing-up theorems of Fujita type are established and the critical Fujita exponent is finite in the not strong degeneracy case, while infinite in the other case. Furthermore, the behavior of solutions at the degenerate point is studied, and it is shown that for the nontrivial initial datum vanishing at the degenerate point, the solution always vanishes at the degenerate point if the degeneracy is strong enough, while never if it is not.

Fujita type theorems for a class of nonlinear diffusion equations

This paper studies a class of weighted non-Newtonian filtration equations with slow diffusion. By using the method introduced by Galaktionov and Levine for the classical non-Newtonian filtration equation, we establish the blow-up theorems of Fujiita type for the extended model, where more difficult and complicated estimates are required to treat the additional degeneracy and singularity. In particular, we prove via a delicate analysis that the critical situation of $p=p_c$ belongs to the blow-up case. The conclusions of this paper quantitatively show the influence of the degeneracy and singularity to the critical Fujita exponents of non-Newtonian filtration equations.

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u t = u p Δu + a(x)u q in ℝ n × (0, T), where p ≥ 1, q > 1, and the positive weight function a(x) is of the order |x| m with m > −2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.

Critical Fujita exponent for the diffusion equation in transparent media with source having power-like non-linearities

In this paper we study blow-up phenomena of the solutions of a reaction-diffusion equation for which the diffusion term corresponds to the diffusion equation in transparent media. We show that for this equation the critical Fujita exponent is smaller than the classical one.

CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS

In this paper, we consider the positive solution to a Cauchy problem in <TEX>$\mathbb{B}^N$</TEX> of the fast diffusive equation: <TEX>${\mid}x{\mid}^mu_t={div}(\mid{\nabla}u{\mid}^{p-2}{\nabla}u)+{\mid}x{\mid}^nu^q$</TEX>, with nontrivial, nonnegative initial data. Here <TEX>$\frac{2N+m}{N+m+1}$</TEX> < <TEX>$p$</TEX> < 2, <TEX>$q$</TEX> > 1 and 0 < <TEX>$m{\leq}n$</TEX> < <TEX>$qm+N(q-1)$</TEX>. We prove that <TEX>$q_c=p-1{\frac{p+n}{N+m}}$</TEX> is the critical Fujita exponent. That is, if 1 < <TEX>$q{\leq}q_c$</TEX>, then every positive solution blows up in finite time, but for <TEX>$q$</TEX> > <TEX>$q_c$</TEX>, there exist both global and non-global solutions to the problem.

Critical exponents for the heat conduction equation with a nonlinear boundary condition

We consider the heat conduction equation ut = ∇·(|∇u| m−1 ∇u )( m> 1),x∈ R N ,t >0, subject to the nonlinear boundary condition −|∇u| m−1 ∂u ∂x1 = u p ,x1 =0 ,t >0, in multi-dimension. We obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions. Mathematics Subject Classification: 35B33; 35K20; 35K57

A reaction–diffusion system with mixed-type coupling

This paper deals with asymptotic behavior of solutions to a reaction–diffusion system coupled via localized and local sources: ut=Δu+vp(x∗(t),t),vt=Δv+uq. Both the initial-boundary problem with null Dirichlet boundary condition and the Cauchy problem are considered to study the interaction between the two kinds of sources. For the initial-boundary problem we prove that the nonglobal solutions blow up everywhere in the bounded domain with uniform blow-up profiles. In addition, it is interesting to observe that the Cauchy problem admits an infinity Fujita exponent, namely, the solutions blow up under any nontrivial and nonnegative initial data whenever pq>1. All these imply that the blow-up behavior of solutions is governed by the localized source for the two problems with mixed-type coupling.

The critical Fujita exponent for the fast diffusion equation with potential

This paper studies the Cauchy problem for positive solutions of the fast diffusion equation with source and quadratically decaying potential ut=Δum−V(x)um+up in Rn×(0,T), where 1−2mα+n<m<1, p>1, n≥2, V(x)∼ω|x|2 with ω≥0 as |x|→∞, and α is the positive root of mα(mα+n−2)−ω=0. We obtain the critical Fujita exponent pc=m+2mα+n to the problem in the sense that every nontrivial solution blows up in finite time when 1<p≤pc, and there are both global and non-global solutions if p>pc.

Critical exponent for the semilinear wave equation with time-dependent damping

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{ll} u_{tt} - \Delta u + b(t)u_t=|u|^{\rho}, & (t,x) \in \mathbb{R}^+ \times \mathbb{R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x), & x \in \mathbb{R}^N. \end{array}\right. (*) $$ When $b(t)=b_0(t+1)^{-\beta}$ with $b_0>0$ and $-1 < \beta <1$ and $\int_{{\bf R}^N} u_i(x)\,dx >0\,(i=0,1)$, we show that the time-global solution of ($*$) does not exist provided that $1<\rho \leq \rho_F(N):= 1+2/N$ (Fujita exponent). On the other hand, when $\rho_F(N)<\rho<\frac{N+2}{[N-2]_+}:= \left\{ \begin{array}{ll} \infty & (N=1,2), (N+2)/(N-2) & (N \ge 3), \end{array} \right.$ the small data global existence of solution has been recently proved in [K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that $0 \le \beta<1$. We can prove the small data global existence even if $-1<\beta<0$. Thus, we conclude that the Fujita exponent $\rho_F(N)$ is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109--114].