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

Abstract

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.

Fujita type theorem for a class of coupled quasilinear convection–diffusion equations

“…where C 1 > 0 is a positive constant independent of l. Substituting ( 22) and ( 23) into (21) shows that…”

Blow-up for a non-linear stable non-Gaussian process in fractional time

“…was investigated by Fujita in 1966. Since then, many other researchers have explored blow-up phenomena (see, e.g., [9], [14], [16], [21]). Following the analysis of this phenomenon, in this work we show that the nonlinearity of (1) leads to the blow-up of positive solutions in a finite time.…”

Fujita-type theorems for a class of coupled semilinear parabolic systems with gradient terms

“…Later, the fact that the critical case p = p c belongs to the blowup case was shown in [2][3][4]. From then on, many mathematicians have focused on the extensions of Fujita's results (see, e.g., [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein). Studies on equations with a gradient term are relatively rich.…”

“…Also, it was shown in [22] that the critical Fujita exponent to the Cauchy problem for (1.7) with b(x) = b(|x|)x is still (1.8), where b satisfies (1.4) and (1.5). A more general case that the coefficients of the derivative of u with respect to time t and source term depend on spatial position was considered in [23]. For more studies about the quasilinear equations with gradient terms, one can see [14,18,20], etc.…”

“…The difference between (1.12) and (1.9) shows that the asymptotic behavior of the coefficients of the gradient term and the exponents in the coefficients of source terms can affect the properties of solutions essentially. The method used in the paper is mainly inspired by [16,18,20,22,23]. To prove the blow-up of solutions, we determine the interactions between the diffusion terms and the gradient terms by energy estimates instead of the pointwise comparison principle.…”