The Research On The Dynamical Behaviours Of Some Nonlinear Systems

This paper is devoted to the study of the dynamical behaviours of several nonlinear systems.We divide this paper into four chapters.In Chapter 1,we introduce the background of the problems considered in this paper and briefly recall some works on related subjects.In Chapter 2,we consider the initial-boundary value problem of a nonlinear damped viscoelastic wave equation where ? is a bounded domain in Rn with a boundary(?)? smooth enough,R+ ={t |0<t<?},? denotes the Laplace operator in Rn and' means The following assumptions on g and h are required.(H.h)is increasing in the second variable,and there exists a strictlyincreasing odd function k ?C1(R+)satisfying is strictly convex on[0,r02]for some constant r0<1,such thatc1a(x)|v|?|h(x,v)|?c2a(x)|v|,(?)x??,if |v|?-1,c3a(x)k(|v|)?|h(x,v)|?c4a(x)k-1(|v|),(?)x??,if |v|?-1,hold for some positive function a(x)? L?(?)and positive constants c1,c2,c 3,c 4,where k-1 denotes the inverse function of k;(H.g)g(t)>0 is a nonincreasing function satisfying?0+?g(?)d?<1,and there exists a nonnegative strictly convex function G? C1(R+)with G(0)= G'(0)= 0,such that g'(t)?-G(g(t))and G(v)?K(v),v?[0,r02].Under these assumptions,we obtain the following theorem about the existence of the solution.Theorem 1.Assume that(H.g),(H.h)hold and u0?H01(?)?H2(?),u1?H01(?).Then the initial-boundary value problem has a unique weak solution u such that Moreover,we use energy method and the Gronwall's inequality to obtain the fol-lowing formula for the energy decay rates.Theorem 2.Let assumptions(H.g)and(H.h)hold,and let u0,u1 be as in Theorem 1.Then we have E(t)?C?(E(0))?(t),(?)t?0 holds for some constant C>0,where E(t)is defined by and ? is a function decaying to 0 as ? +? satisfying the functions L,v and the constant C11 are defined in(2.47),(2.50)and(2.49)respectively.The above theorem gives a formula for the energy estimate of problem(3).It is well known that both damping term and viscoelastic term can cause energy decay,but there are few works concerning about the case when both two types of terms exist.Though considered in some works,the damping term and viscoelastic term are often linear.In our work,we deal with more complicated cases.Moreover,we compare the decay rates caused by the two terms and discover the relation between them.The formula in Theorem 2 is suitable for many types of viscoelastic wave equations with nonlinear damping.At the end of Chapter 2,we will present the applications of our results to two classes of specific examples and give the energy decay rates of these systems.In Chapter 3,we consider the following initial-boundary value problem of a nonlinear damped beam equation with variable coefficients.where ?(?)Rn is a nonempty bounded open set with a C2 boundary(?)?,R+={t|0<t<+?},?·? is the L2 norm in Rn,? and ? denote the gradient operator and the Laplace operator in Rn respectively,' means(?),v is the exterior unit normal vector to the boundaryac(?)? and(?)represents the normal derivative.Beam equations with variable coefficients can be used to describe the vibration of a inhomogeneous beam.Compared to beam equations with constant coefficients,they are more effective when dealing with real models.At the same time,a lot of problems are raised from the existence of variable coefficients.In our work,we define two special functionals and use energy method to overcome these problems.Such method is useful for both linear and nonlinear damped cases.In our work we require the terms a(x,t),b(x,t),M(x,t,?),h(s)satisfy the fol-lowing assumptions.Under these assumptions,we obtain the following theorems on the existence and uniqueness of the global strong solution.Theorem 3.(existence)Assume that(H.a),(H.b),(H.M)and(H.h)hold,and u0?H02(?)?H4(?)u1 ? H02(?),b0 is a constant large enough such that and where k1=2?1a1-2(2A0+?22),C is the Poincare constant satisfying?u??C??u?,??u??C??u?,?u??C??u?,for any u ? H02(?),and the function H(t)and constant C2 will be given in(3.12)and(3.17)respectively.Then the problem has at least one strong solution u(x,t).Theorem 4.(uniqueness)Assume(H.a),(H.b),(H.M)and(H.h)hold,and also assume thatM ?C1((?×[[0,+?))×[0,?)),?M,?M ?L?[?×[0,+?)×[0,+?)).Then the global strong solution of the problem above is unique.Moreover,we also prove that the energy E(t)of the above system decays ex-ponentially.Theorem 5.Under the assumptions of Theorem 4,there exist constants ?>0 and ?>0 such that the energy E(t)of the problem satisfies E(t)??e-?t,(?)t ? 0,where E(t)is defined by E(t)=1/2??(a(x,t)|u'|2 + b(x,t)|?u|2 + M(x,t,??u(t)?2)|?u|2)dx.In Chapter 4,we consider the following Poisson-Nernst-Planck system.where x ?(0,1),k ? 1,2,...,n,and the boundary value conditions are?(0)= V,ck=(0)=lk?0;?(1)=0,ck(1)= rk ? 0.where the unknown variables are the electric potential ?,the concentration(number density)ck and the flux density Jk of the kth ion species.The interval[0,1]is the scaled one-dimensional ion channel with x = 0 and x = 1 representing the two open ends of the channel,?2?1 is a dimensionless parameter,h(x)represents the cross-section area of the ion channel over x,Q(x)is the permanent.charge,and,for the kth ion species,?k?0 0 is its valence,lk and rk are its concentrations at the boundaries.Under the definitions of the following variables we consider the distribution of the eigenvalues ?2,?3 of a matrix D corresponding to the PNP system when n = 3.We transfer the problem of finding the eigenvalues of D into finding the zeros of h(z)and obtain the following results.Proposition 1.When ml = mr,one has which has two rootsWhen mr<ml,h(z)has two distinct real roots ?2 and ?3.More precisely,A3>ml andProposition 2.When m,>ml and pmr<Ml,h(z)always has two real roots?2 and ?3 with ?2<0<?3 ml.When mr>ml and pmr = ml,h(z)always has two real roots ?2 and ?3 withTheorem 6.When mr>ml and pmr>Ml,we have the following cases.and there is a negative double root;then and there is a double root in the interval(mr+ml,?).and there is a positive double root in the interval(0,ml);in particular,p ?when then and there is a double root in the interval(mr,mr+ml);then the double root is mr+ ml.has a pair of complex conjugate roots.has two positive real roots ?2,?3>mr.has two real roots of same sign:two negative real roots for and two positive roots inBase on these results on the distribution of the eigenvalues,we will provide the formulas of the current L in different conditions.Then,we use the formulas to analyse the relation between the current and the voltage.