Blow-up Of Solutions To Some Nonlinear Evolutional Equations

Along with the rapid development of modern science and technology in the fields of physics, chemistry, biology and engineering science, etc., many fields of science are continuously put forward a lot of mathematical models, and many of them are widely involved in some evolutional parabolic equa-tions. In most of the cases, the equations involved are nonlinear and possess degeneracy or singularity. In the recent decades, many scholars have made significant progress in studying such models.This paper mainly studies some problems on some evolutional equations with double degeneracy, a porous medium equation with nonlocal boundary conditions and a moving localized source, and some nonlinear parabolic equations with time delay under dynamical boundary conditions. The topics include the effect of the nonlinear source. nonlocal boundary conditions, the moving localized source, the nonlocal source and coupling among them on the critical exponents of blowing up of the solutions. This thesis consist of three chapters, the main contents are as the following:In chapter 1, we investigate the global existence and blow-up property of positive solutions for the following Cauchy problem of non-Newtonian polytropic filtration equations where p>1,q>max{1,m(p-1)},m≥1,and both w(x)≠0 and u0(x) are nonnegative functions in Rn.The main interests of this paper are to study the large time behavior of positive solutions of(1). In the case m(p-1)≥1,the equation has degeneracy or singularity,namely)if m> 1 or p> 2,the equation has degeneracy at the points where u(x,t)=0 or▏▽u(x,t)▕=0,and if 0< m<1 or 11,q>max{1,m(p-1)},m≥1,and both w(x)(?)0 and u0(x)are nonnegative functions in Rn. If n≤p,then every positive solution of(1)blows up in finite time.In Chapter 2, we study the positive solutions of the porous medium equa-tions with nonlocal boundary conditions and a moving localized source where m>1 is a constant andΩis a bounded domain in RN(N>1)with smooth boundary (?)Ω.k(x,y)(?)0 is a nonnegative continuous function defined for x∈(?)Ωand y∈Ω-,while u0(x)is a positive continuous function and satisfies the compatibility condition u0(x)=∫Ωk(x,y)u0(y)dy for x∈(?)Ω.x0(t)is a continuously differentiable function from R+ to K,a fixed compact subset ofΩ.f(s)satisfies the assumptions as follows: (H1) f(s)∈C[0,∞)∩C1(0,∞);(H2) f(0)≥0 and f'(s)>0 in (0,∞).We first give the definitions of the subsolution and supersolution for the problem (2).Definition 2 A function u is called a subsolution to Problem (2) on QT, if u∈C2,1(QT)∩C(QT∪ΓT) satisfiesA supersolution is defined in a similar way with each inequality reversed.We established the comparison principle about problem (2), and then we proved the global existence and blow-up of the classical solution for (2).Theorem 3 Let u and v be a nonnegative subsolution and a superso-lution to (2) respectively, with u(x,0)≤v(x,0) for x∈Ω. Then u≤v in QT if either v≥δor u≥δholds for someδ> 0.Compared with the homogeneous Dirichlet boundary condition, the weight function k(x,y) plays an important role in the global existence or blow-up results for Problem (2).The main results of this chapter are as follows:Theorem 4 Assume that∫Ωk(x,y)dy=1 for x∈(?)Ωand f(s) satisfies (H1) and (H2). Then every solution to (2) exists globally when∫s0∞(ds)/(f(s))=∞while it blows up in finite time when∫s0∞(ds)/(f(s))<∞for some s0>0. Theorem 5 Assume that∫Ωk(x,y)dy>1 for x∈(?)Ω.Then the solution to(2)blows up in finite time provided that∫s0∞(ds)/(f(s))<∞.Theorem 6 Assume that∫Ωk(x,y)dy＜1 for x∈(?)Ωand f(s)satisfies (H1)and(H2).(1)Then every solution to(2)exists globally whenthen the solution to(2)exists globally when u0(x) is suitably small.To prove the blow-up results in the case of∫Ωk(x,y)dy<1,we need a additional assumption as follows:(H1')f(s)∈C1[0,∞),(f'(s))/(sm-1) is nondecreasing in(0,∞)and∫s0∞(sm-1)/(f(s))ds< +∞for some s0>0.Obviously,it is easy to find the functions f(s)that satisfying the(H1'), for example,the exponential function f(s)=es or the f(s)=sp(p>m).Theorem 7 Assume that f(s) satisfies(H1')and(H2).Then the unique solution u(x,t)to Problem(2)blows up in finite time if u0(x)is sufficiently large.At the end of this chapter,we proved that the blow-up is global for the problem(2).Definition 3 Suppose that u(x,t)blows up in a finite time T.We say that x* is a blow-up point of u(x,t)ifWe say that the blow-up is global if every point inΩis a blow-up point. So we get:Theorem 8 If the solution u(x,t)to Problem(2)blows up in a finite time T,then u(x,t)blows up globally.In Chapter 3, we consider the blow up property for a nonlinear parabolic equation with time delay under dynamical boundary conditionswith 1≤k< p∈R, m∈N, m≥2, and the problemwith 2k+1< p∈R,k, m∈N.Assume thatΩC Rn is a bounded domain, whose boundary aΩis of class C2. On the time lateral boundary, we impose some dynamical conditions, involving the outward normal derivative and the time derivative. The unit outward normal vector field and the outward normal derivative are respec-tively denoted by v:aΩ→Rn and av. The dissipativity condition isWhen study the classical solutions, we assume thatAs for the initial data, we require The main results are the following:Theorem 9 Suppose p>k≥m+1,if one of the following conditions is satisfied: then the classical positive solution u(x,t) of Problem (3) blows up in a finite time T which satisfiesRemark 1 This result can be extended to the equation with Neumann boundary conditions.Remark 2 In the equations, if we replaced exp(pu) with f(u)≥M exp(ku), the conclusion still holds.Theorem 10 Soppose p> 2k+ 1, then the classical positive solution u(x,t) of Problem (4) blows up in finite time T which satisfiesRemark 3 This result can also be extended to the case f(u)≥a▕u▕p.