The Fujita Exponent Of Semilinear Parabolic Equations With Variable Exponent Reactions

In this thesis, we consider the following semilinear parabolic problem with a variable exponent reaction under Robin boundary conditions where Ωc is a bounded domain of RN, and Ω is an exterior domain of RN. We suppose that if N≥2, the boundary (?)Ω is of class C2. ν:(?)Ω-RN is the unit outward normal vector and (?)ν is the outward normal derivative. p(x)∈C(Ω) is a bounded function satisfying p(x)>1, and we denote p_=infx∈Ωp(x), p+=supx∈Ωp(x). a is a nonnegative continuous function defined on (?)Ω x (0,∞), and the initial data φ verifies φ∈C(Ω),0<‖φ‖∞<∞,φ>0. In this thesis, we only consider the positive classical solutions of the above problem, i.e. u(x,t) is a positive solution of the problem and u(x,t)∈C(Ω×[0,∞))∩C2,1(Ω×(0,∞)).This thesis mainly concerns with the Fujita phenomenon of the above semilinear parabolic problem with variable exponent reactions under Robin boundary conditions, and the main results are the following:(1) If p->1+(2/N), then the above problem admits global nontrivial solutions provided the initial datum φ(x) is small enough;(2) If1<p-≤p+<1+(2/N), then all the nonnegative nontrivial solutions of the above problem blow up in finite time.This thesis consists of3chapters.In the first chapter, we briefly recall the background of the problem under consideration and state the main results of this thesis as well as some prospects of further development of some related problems. The second chapter is the main part of this thesis and we mainly study the Fujita critical exponent of the above problem. In the first part of this chapter, we give the definition of positive solutions, lower and upper solutions as well as the comparison principles and maximum principles that will be used in the proof of the main results. Moreover, since the problem we study is in exterior domains, we also sketch the outline of the proof of the local existence of classical solutions by using the truncation method. In the second part of this chapter we show that Fujita phenomenon can still happen for our problem. More precisely, we can prove, by constructing a global super-solution when the initial data are suitably small, that the problem admits global solutions if p->1+(2/N). If1<p-≤p+<1+(2/N) by constructing lower solutions which can be proved to blow up in finite time with Kaplan’s method and Jensen inequality, we can show that all the nonnegative nontrivial solutions of the above problem blow up in finite time.In the third chapter, we summarize the work of this thesis and propose some possible research prospects in this field.