Two Types Of Nonlinear Wave Equations Definite Solution Of The Problem

In this paper, we study the existence, the uniqueness of the local generalizedsolutions, the global generalized solutions, the global classical solutions, the decayrate of the global solutions for the initial boundary value problems and the Cauchyproblems to some nonlinear evolution equations, and give the sufficient conditionsof blowup of the solutions for some problems above. The main results include thefollowing four parts:In Chapter 2, by virtue of the Green function of the ordinary differential equation, the contraction mapping principle and the extension theroem of solutions, weprove the existence, the uniqueness and the regularities of the global generalizedsolution and the global classical solutions for the initial boundary value problemsfor a class of generalized IMBq equations with damping termAnd we give the sufficient conditions of blowup of the solution for the problem(1)-(3). The main results are as follows:Theorem 1 Suppose that u₀(x),u₁(x)∈C³[0,1],u₀(0) = u₀(1) = u₁(0) =u₁(1) =0, f∈C³(R), g∈C¹(R), |f'(s)|,|g'(s)|≤C₁, then the problem (1)-(3) hasa unique global classical solution u∈C²([0, T]; C²[0, 1]), (?)T > 0.Theorem 2 Suppose that u₀(x),u₁(x)∈C³[0,1],u₀(0) = u₀(1) = u₁(0) =u₁(1) = 0, g = f∈C³(R), and where F(s) = (?), A,B > 0 are all constants. Then the problem (1)-(3) has aunique global classical solution u∈C²([0, T]; C²[0, 1]), (?)T > 0.Theorem 3 Assume that u₀(x),u₁(x)∈C³[0,1],u₀(0) = u₀(1) = u₁(0) =u₁(1) = 0, g = 0,αβ+γ= 0, f∈C³(R), (4) holds andwhere C, D > 0 are constants. Then the problem (1)-(3) has a unique global classicalsolution u∈C²([0, T]; C²[0,1]), (?)T > 0.Theorem 4 Assume that u(x, t) is the classical solution of the problem (1)-(3),if the following conditions hold: . . .whereλ=Ï€², (3) (?) grows fast enough, such thatconverges whenαβ-λγ≤0. Then whenαβ-λγ> 0,for some finite time t₀≤T₂ = (?); whenαβ-λγ≤0 ,for some finite time t₀≤T₁. In Chapter 3, by using of the fundamental solution of the ordinary differentialequation, the contraction mapping principle and the extension theorem of solutions,we prove the existence, the uniqueness and the regularities of the global generalizedsolutions and the global classical solutions in W^k,p(R) for the Cauchy problems fora class of generalized IMBq equation with damping termMoreover , we show the decay rate of the solutions for the problem (5), (6), andgive the sufficient conditions of blowup of the solution for the problem in finite time.The main results are stated as followsTheorem 5 If u₀, u₁∈W^k+2,p(R),f,g∈C^k+3(R), where k > 1/p, and(?)_s∈R, |f'(s)|, |g'(s)|≤A₀. Then the problem (5), (6) admits a unique global clas-sical solution u∈C³([0,∞); W^k+2,p(R)), that is, u∈C³([0,∞);C²(R)∩L^∞(R)).Theorem 6 If u₀, u₁∈W^k+2,p(R),g = f∈C^k+3(R), where k > 1/p, F(s)≤0or f'(s)≤A₀. Then the problem (5), (6) has a unique global classical solutionu∈C³([0,∞);W^k+2,p(R)), that is, u∈C³([0,∞);C²(R)∩L^∞(R)).Theorem 7 Suppose that u₀, u₁∈W^k+2,p(R)∩L²(R),∧^-1u₁∈L²(R), g = 0,β= 0, f∈C^k+3(R), where k>1/p is nonnegative integral number, and F(u₀)∈L¹(R), f'(s)≤A₀ or F(s)≤0 satisfyingwhere A, B > 0 are constants,∧^-rΨ=F^-1 (|x|^-rFΨ(x)),F,F^-1 denote the Fouriertransform and the inverse transform. Then the problem (6), (7) admits a uniqueglobal classical solution u(x, t)∈C³([0,∞); W^k+2,p(R)∩L²(R)), that is,u∈C³([O,∞);C²(R)∩L²(R)∩L^∞(R)).Theorem 8 Suppose thatβ= 0,γ≤0,g(s) = 0, f(s)∈C(R),F(s) = (?).u₀, u₁∈L²(R),∧^-1u₀,∧^-1u₁∈L²(R),F(u₀)∈L¹(R), there existδ₁,δ₂> 0, such thatThen the solution of the problem (5), (6) blows up in finite time if one of the followin9conditions holdswhereTheorem 9 Suppose thatβ= 0,γ≤0,g(s) = 0,f(s)∈C(R),F(s) = (?),u₀, u₁∈L²(R),∧^-1u₀,∧^-1u₁∈L²(R), F(u₀)∈L¹(R) and there existδ₁,δ₂ > 0,such thatThen when (?)(0)≤0, andthe solution of the problem (5), (6) blows up in .finite time.In Chapter 4, by using of the Fourier transform , the contraction mappingprinciple and the extension theorem of solutions, we study the existence and the uniqueness of the global generalized solutions and the global classical solutions inH^s(R) for the Cauchy problem (5), (6) are studied. The main results are stated asfollowsTheorem 10 If u₀, u₁∈H^s(R), g = f∈C^[s]+1(R),s > 5/2, F(s)≤0, orf'(s)≤A₀. Then the problem (5), (6) admits a unique global classical solutionu(x,t)∈([0,∞);C²(R)).Theorem 11 If u₀, u₁∈H^s(R),g = 0, f∈C^[s]+1(R), and s > 5/2.∧^-1u₁∈L²(R), and F(u₀)∈L¹(R), f'(s)≤A₀ or F(s)≤0 satisfyingwhere A, B > 0 are all constants. Then the problem (5), (6) has a unique globalclassical solution u∈C²([0,∞); C_B¹(R)).In Chapter 5, we study the energy decay of the following problem of viscoelasticity equation with nonlinear damping on the boundaryBy virtue of a so-called energy perturbation method and a comparison inequality, we prove that the solution of the problem (8)-(10) and f(x, t) have the sameexponential decay or algebraic decay for the case F(s) = s + |s|^α-2(α≥2) andF(s) = |s|^α-2(α>2). The main results are as follows:Theorem 12 (1) Suppose thatwhere M₁≥0,λ₁ > 1, then there exists C₄ > 0, such that (2)Assume thatwhere M₂≥0,λ₂>0, then there exists C₅ > 0, such thatwhere (?) = min(λ₂, C₁).Theorem 13 If there exist M₃≥0,λ₄ > 1, such thatthen (?)t∈[0,∞), there exist C₆, C₇ > 0, such that...