Critical Fujita Exponent Research Articles

Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux

In this paper, we deal with a fast diffusive polytropic filtration equation ∂ u ∂ t = ∂ ∂ x ( | ∂ u m ∂ x | p − 2 ∂ u m ∂ x ) ( 1 < p < 1 + 1 m ) in R + × ( 0 , + ∞ ) , subject to a nonlinear boundary flux − | ∂ u m ∂ x | p − 2 ∂ u m ∂ x ( 0 , t ) = u q ( 0 , t ) , t ∈ ( 0 , + ∞ ) . 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 polytropic filtration equation, furthermore give the blow-up set and upper bound of the blow-up rate for the nonglobal solutions.

Critical exponent for non‐Newtonian filtration equation with homogeneous Neumann boundary data

AbstractThis paper is concerned with large time behavior of solutions to the homogeneous Neumann problem of the non‐Newtonian filtration equation. It is shown that the critical Fujita exponent for the problem considered is determined not only by the spatial dimension and the nonlinearity exponent, but also by the coefficient k of the first‐order term. In fact, we show that there exist two thresholds k∞ and k1 on the coefficient k of the first‐order term, and the critical Fujita exponent is a finite number when k is between k∞ and k1, while the critical exponent does not exist when k⩽k∞ or k⩾k1. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources

This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depends on the spacial dimension. Exactly speaking, the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this threshold.

Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data

In this paper, we establish the blow-up theorems of the Fujita type for a class of homogeneous Neumann problems of quasilinear equations with convection terms. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited such that the critical Fujita exponent even could be infinite for the model considered in the paper owing to the effect of convection.

An equation whose Fujita critical exponent is not given by scaling

In this paper, we prove sharp blow up and global existence results for a heat equation with nonlinear memory. It turns out that the Fujita critical exponent is not the one which would be predicted from the scaling properties of the equation.

Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition

This work is concerned with the critical exponent of the non-Newtonian polytropic filtration equation with nonlinear boundary conditions. We obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions.

Critical Fujita exponents of degenerate and singular parabolic equations

In this paper we investigate the critical Fujita exponent for the initial-value problem of the degenerate and singular nonlinear parabolic equation with a non-negative initial value, where p &gt; m ≥ 1 and 0 ≤ λ1 ≤ λ2 &lt; p(λ1 + 1) − 1. We prove that, for m &lt; p ≤ pc = m + (2 + λ2)/(n + λ1), every non-trivial solution blows up in finite time, while, for p &gt; pc, there exist both global and non-global solutions to the pro

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u t - ▵ u = | u | p - 1 u in R N . The initial data is of the form u 0 = λ ϕ , where ϕ ∈ C 0 ( R N ) is fixed and λ > 0 . We first take 1 < p < p f , where p f is the Fujita critical exponent, and ϕ ∈ C 0 ( R N ) ∩ L 1 ( R N ) with nonzero mean. We show that u ( t ) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1 < p < p s , where p s is the Sobolev critical exponent, and ϕ ( x ) decaying as | x | - σ at infinity, where p < 1 + 2 / σ . We also prove that u ( t ) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ϕ is not radial.

Decay estimates for the damped wave equation with weighted odd initial data

In this paper, we consider the Cauchy problem to the damped wave equation. Several authors have shown the diffusive structure of the problem which has been represented by L p – L q estimates. We show the new L p – L q estimates to the problem when initial data are odd and belong to certain weighted function space. These L p – L q estimates imply the new decay estimates for the above Cauchy problem. Moreover, we apply the above decay estimates to the Cauchy problem for the semilinear wave equation with power nonlinearities lower than Fujita critical exponent. We show the existence of the time global solutions when the initial data are small and odd.

Existence and blow-up for higher-order semilinear parabolic equations: Majorizing order-preserving operators

As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = -(-Δ) m u + |u| P , x ∈ R n , t > 0; u(x, 0) = u 0 (x), x ∈ R n , where m > 1, p > 1, with bounded integrable initial data u 0 , the critical Fujita exponent is p F = 1 +2m/N, so that for p > p F there exists a class of small global solutions, and for p ∈ (1, p F ] blow-up can occur for arbitrarily small initial data. The analysis of the asymptotics of both classes of global and blow-up solutions is based on comparison with similarity solutions of the majorizing order-preserving integral equation, which is shown to exist for any m > 1. Generalizations to differential and integral evolution equations and relations to positivity sets for higher-order equations are discussed.

Critical Exponent for a Nonlinear Wave Equation with Damping

Abstract In this study we solve the critical exponent problem for the nonlinear wave equation with damping. We show that this critical exponent coincides with the famous Fujita critical exponent for the heat equation. We also observe a futher interesting phenomenon due to the presence of the damping term. The “real” support of the solution is strongly suppressed by the damping and is concentrated in the ball |x| 0 the solution decays exponentially.

Critical exponent for a nonlinear wave equation with damping

Abstract In this study we solve the critical exponent problem for the nonlinear wave equation with damping. We show that this critical exponent coincides with the famous Fujita critical exponent for the heat equation. We also observe a futher interesting phenomenon due to the presence of the damping term. The “real” support of the solution is strongly suppressed by the damping and is concentrated in the ball |x| 0 the solution decays exponentially.