Cauchy Problems For Two Classes Of Nonlinear Evolution Equations

This paper consists of five chapters. The first chapter is the introduction. In the second chapter, we will study the existence and uniqueness of the local generalized solution and the local classical solution to the Cauchy problem for the damped nonlinear hyperbolic equation. In the third chapter, we will prove the blow-up of the solution to the problem mentioned in Chapter two and give an example. In the fourth chapter, we will discuss the existence and uniqueness of the global generalized solution and the global classical solution to the Cauchy problem for Bq model equation. In Chapter five, we will prove the blow-up of the solution to the problem mentioned in Chapter four and give an example.In the second chapter, we study the following Cauchy problem for the damped nonlinear hyperbolic equationwhere u(x, t) denotes the unknown function, k1 and k2 are two positive constants, V denotes the gradient operator, (?)² = â–³ denotes Laplacian operator, V4 = A2 denotes the biharmonic operator, g(s) is the given nonlinear function, u0(x) and u1(x) are given initial value functions, and subscript t indicates the partial derivative with respect to t. For this purpose, we first consider the periodic boundary value problem of the equation (1)After the existence and uniqueness of the local generalized solution and the local classical solution to the problem (1), (3), (4) are proved, using the sequence of the periodic boundary value problems we prove that the Cauchy problem (1), (2) has a unique local generalized solution and a unique local classical solution. The main results are the following:Theorem 1 Suppose that g ∈ C⁴(R), |g(s)| ≤ K1|s|q, |g'(s)| ≤ K2|s|q-1 etc., where q > 2 is a natural number and K1, K2 are positive constants. If u0 ∈ H⁶(Ω) and u1 ∈ H⁴(Ω), then the periodic boundary value problem (1.1), (2.1), (2.2) admits a unique local generalized solution u(x, t).Theorem 2 Suppose that g ∈ C¹0(R), |g{s)| ≤L₁|s|^q, |g'(s)| ≤ K2|s|q-1 etc., where q ≥ 2 is a natural number and K1, K2 are positive constants. If u0 ∈ H¹2(Ω) and u1 ∈ H¹0(Ω), then the periodic boundary value problem (1.1), (2.1), (2.2) admits a unique local classical solution u(x,t).Theorem 3 Suppose that g G C4(R), \g(s)\ < K^s]*, \g'(s)\ < K2\s\q¹ etc., where q > 2 is a natural number and Kj_, K2 are positive constants. If u0 G H6(R3) and tti € H4(R3), then the Cauchy problem (1.1), (1.2) admits a unique local generalized solution u(x,t). If g G CW(R), uQ G Hn(R3) and m G H10(R3), then the Cauchy problem (1.1), (1.2) admits a unique local classical solution u(x, t).In Chapter three, the blow-up of the solution to the problem (1), (2) are proved by means of the concavity method. And we give an example. The main results are the following:Theorem 4 Suppose that u0 G #2(fl3), ux G L2(R3), G(Au0) G L^R3) and there exists a constant j3 > 0 such thatsg(s) < 2(2/? + l)G(s) + 2/%is2, Vs G R,where G(s) = /os g(r)dT.Then the generalized solution u(x,t) or the classical solution u(x,t) of the Cauchy problem (1), (2) blows-up in finite time if one of the following conditions holds:(1) E(0) < 0;(2) E(0) = 0, and 2(3 jR3 uoUldx - fc2||Au0||2 > 0;(3) ï¿¡7(0) > 0, JR3 uoiixdx > 0 and4WR3UoMz)2-4/?2ï¿¡(O)NI|2-||Mwhere ï¿¡7(0) = ||ui||2 + fci||Auo||2 + 2 JR3 G(Auo)dx.In the fourth chapter, we will discuss the following Cauchy problem for Bq model equationutt - uxx - uxxtt - [iuxxxx + uxxxxtt = f(u)xx, x 6 R, t>0, (5)u(x,0) = uo{x), ut(x,0) = u1(x), x e R, (6)where u(x, t) denotes the unknown function, // < 0 is a constant, f(s) is the given nonlinear function, uo(x) and u\(x) are given initial value functions, and subscript x and t indicate the partial derivative with respect to x and L respectively. By the change u(x, t) — vx(x, t),rx rx{x) = I uo(y)dy, ip(x) = / ui(y)dy, we get?/—oo J^ooVxtt vxxx - vxxxtt - fivx5 + vxHt = f(vx)xx, x E R, t>0, (7)v(x,0) =Moreover, we can get the existence and uniqueness of the global generalized solution and the global classical solution of the problem (9), (10). By the change u(x,t) = vx(x,t), uQ(x) = x(x), ui(x) — ipx(x) , we can prove the existence and uniqueness of the global generalized solution and the global classical solution of the problem (5), (6). The main results are the following:Theorem 5 Suppose that / e Ck+1(R) and there is a constant Co < 0, such that f'(s) > Co for any s G R, , ip 6 Hk+2{Q)(k > 0 is a natural number). If k > 2, then Cauchy problem (9), (11), (12) admits a unique global generalized solution v(x,t), which has continuous derivatives vx*tr(x, t)(0 < s < k,r = 0,1, 2) and generalized derivatives vxHr(x,t)(0 < s < k + 2,r = 0,1,2,3). If k > 4, the Cauchy problem (9), (11), (12) admits a unique global classical solution v(x,t) , which has continuous derivatives vx?tr(x, t)(0 < s < k, r — 0,1, 2) and generalized derivatives vx*v{x, t){0 < s < k + 2, r — 0,1, 2, 3).Theorem 6 Suppose that / G Ck+1(R) and there is a constant Co < 0, such that f'(s) > Co for any s € R, (f), ip € Hk+2(R). If k > 2, then Cauchy problem (9), (10) admits a unique global generalized solution v(x, t), which has continuous derivatives vx>f{x, i)(0 < s < k,r = 0,1, 2) and generalized derivatives vxstr(x,t)(0 < s < k + 2,r = 0,1,2,3). If k > 4, the Cauchy problem (9), (10) admits a unique global classical solution v(x, t), which has continuous derivatives vx,tr(x,t)(0 < s < k, r = 0,1, 2) and generalized derivatives vx.t'(x,t)(0 < s < fe + 2,r = 0,l,2,3).Theorem 7 Suppose that / G Ck+1(R) and there is a constant Co < 0, such that f'(s) > Co for any s G R, uo(x), u^x) G Hk+1(R). If k > 3, then Cauchy problem (5), (6) admits a unique global generalized solution u(x,t), which has continuous derivatives uxHr(x,t)(0 < s < k — l,r = 0,1,2) and generalized derivatives uxstr(x,t)(0 < s < k + 1, r = 0,1, 2,3). If k > 5, the Cauchy problem (5), (6) admits a unique global classical solution u(x, t), which has continuous derivatives uxstr(x, t)(0 < s < k — 1, r = 0,1, 2) and generalized derivatives ux^r(x, i)(0 < s < k + 1, r = 0,1, 2,3).In the fifth chapter, we first prove the blow-up of the solution to the problem (9),(10). Then we can get some sufficient conditions of blow-up of the solution to the Cauchy problem (5), (6) in finite time. The main results are the following:Theorem 8 Suppose that 4>,ip e H2{R), f e C(R), F(s) = /os f(y)dy, F(s) e L^R) andf(s)s < (4/5 + 2)F(s) + 2/5s2, Vs e R,where /? > 0 is a constant. Then the generalized solution v(x,t) or the classical solution v(x,t) of the Cauchy problem (9), (10) blows-up in finite time if one of the following conditions holds:(1) E(0) < 0;(2) .E'(O) = 0, (...