Asymptotic Behavior Of Solutions To A Degenerate Quasilinear Parabolic Equation With A Gradient Term

Fujita studied the following Cauchy problem at 1966. And creatively proved if 1＜p＜pc= 1+2/n, the Cauchy problem does not have any nontrivial global nonnegative solution, whereas there exist both non-global nonnegative and nontrivial global solution if p＞pc. Since then, we call pc the critical Fujita exponent, and the corresponding conclusions were called blowing-up theorem of Fujita type.Since the blowing-up theorem of Fujita type has been discovered, there have been a number of extensions of Fujita’s results in several directions. Such as different equan-tions, including semilinear parabolic equations and quasilinear parabolic equations; in various of geometries, including whole spaces, cones, exterior domains, nonlinear sources including nonhomogeneous boundary conditions. At present, the blowing-up theorem of Fujita type has got very good extension, even the critical case p= pc is also studied by some mathematics workers.In this thesis, we study the asymptotic behavior of solutions to the following Cauchy problem where p＞m＞1,0≤uo ∈ L∞(Rn), b ∈ C0,1([0,+∞)) satisfies if -n＜≤+∞, thenWe successfully created a blowing-up theorem of Fujita type, proved the critical Pujita exponent to the up Cauchy problem can be formulated as That is to say, if m＜p＜pc, there does not exist any nontrivial nonnegative glob-al solution to the Cauchy problem, whereas if p＞pc, there exist both nontrivial nonnegative global and blow-up solutions to the Cauchy problem.To prove the solutions of Cauchy problem have behavior of blow-up, the methods in this thesis we used that we took the interactions among the gradient, the reaction and the diffusion by a energy integral estimate rather than pointwise comparisons. The key is to choose a suitable energy integral. To prove the existence of global nontriv-ial solutions, we construct a global nontrivial supersolution. Noting that the Cauchy problem does not possess a self-similar construct, we have to seek a complicated super-solution and do some precise calculations. Although (4) is used only for constructing a global nontrivial supersolution and it seems necessary when one constructs such a supersolution.This article is organized as follows. We give some preliminaries in the first chapter, and the definition of (1)-(2) and some auxiliary lemmas are given in the second chapter. In the third chapter, the blow-up theorems of Fujita type for (1)-(2) are obtained.