Large Time Behavior of Solutions to Semilinear Parabolic Equations with Gradient

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

“…In this paper, it is proved that the critical Fujita exponent to the problem can be formulated as p c = m + 2/n. The main methods are inspired by [15,19,20,24,26]. We can prove that there are the blow-up solutions by using the integral estimation method.…”

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].…”

Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms

“…Moreover, [18,21] also investigated the Neumann exterior problems for the Newtonian and non-Newtonian filtration equations with similar gradient terms. [23] showed that the critical Fujita exponent to the Cauchy problem (1.5) with b = b is still (1.6), where b satisfies (1.4). Furthermore, Suzuki [14] considered the Newtonian filtration equation with similar gradient term and got the critical Fujita exponent for some special cases.…”

“…The difference between (1.8) and (1.7) shows that the gradient term can affect the large time behavior of solutions essentially. The technique used in this paper is mainly inspired by [16,18,21,23]. To prove the blow-up of solutions, we determine the interactions among the diffusion and the gradient by a series of precise energy integral estimates instead of pointwise comparison principle.…”