| In this paper we consider initial boundary value problems of the following P-Laplace equationswhere is a bounded domain in Rn with a smooth boundary. We assume that F C1(R+ x x R x Rn, R) satisfies some conditions.In [4], Constantin proved global existence and blows-up of solutions n(f, x) for the following problemThe author shows that if the nonlineanty F(t, x, r, ) satisfies the growth condition\F(t, x, r, )| h(r)(1 + 2), (t, x, r, r,) R+ x x R x Rn, (3)for some function h(r) C(R,R+), it suffices to find an L -a priori bound to guarantee global existence for the solutions of (2). And the author also shows that if (3) deos not hold, it is possible that the solutions of (2) remains uniformly bounded but blows-up in Wk,p( ) in finite times. Take n = 1, a = 1 and F(t, x, r, ) = e , cf. [16].In [6], Constantin use a comparison principle of a ordinary equation z'(t) -w(t, z), z(0) = z0 to deeply reserch global existence of solutions for quasilinear parabolic systems.By studying the former results, we can find that the authors have studied the global existence of solutions in multi-dimensional space, given blows-up results, and the nonlinearity has general cases. Simultaneity we also find that the coefficients only are related with u, v in these problems. In this paper, the coefficients of the operator which the author try to consider are |Vu v~~'*, namely the case is evolution P-Laplace operator. Hereinafter we need to assume that solutions of problem (1) are C2 in x. In the section 4. for systems we need similar conditions.We consider problem (1). if F(t. x. u. Vu) is a locally Lipschitz continuous function, given UQ e H/0fe'p(?) with k 6 [l,min{l -f- !//>, 2 - n/p}), we assume that there exists some maximal T = T(UQ) > 0 and the solutions satisfiesu(t, x} e Cl((0, T); C( )) C((0, T); C2( )) C([0, T); W0k,pIn Section 2, we prove an abstract result which is applied in Section 3 to obtain the global finiteness of solutions to (1) for an appropriate function h. In Section 4, we use two lemmas about an ordinary differential equation to prove the global finiteness of solutions for systems.In order to prove global finiteness of solutions to (1). we first given a theorem as the following:Theorem 1 Let T > 0 and u W1,1((0,T);C( )), where ft is a bounded domain in Rn. Then for every t (0, T) there exists at least one pair of points (t), (t) withm(t) := min[u(f,x)] = u(t,?t)). M(t) := max['u(/..;:)] = u( and the functions m, M are almost everywhere differentiable on (0. T) withBy using this result, we can prove global imiteness of solutions to (1). Theorem 2 Assume that F(*. .r..u, Vu) t Cl(R~ x ii x /{ x /?",/?,) satisfies the growth condition|F(i, x, r, )| < (r) 2 + (r) + r), (i, x, r, ) R+ x x R x Rn,with some a, C(R, R+), C(R, (0, )) such thatThen for any initial data UQ e W0'p( ) the corresponding solutions to (1) is global finiteness in time.After we get the global finiteness results to (1), we prove the similary results to the following system:where fi is a bounded domain in Rn with a smooth boundary. We assume thatgiven UO.V0 W0fc'p(Q). the solutions (u, v) to (4) is defined on interval (O.T) for maximal time T = T(u0,v0) > 0 and (u,v) satisfiesWe first give the following two theorems:Theorem 3 Let w(t, z) C(R+ x R, R) and r(t) be the maximal solu-tion of the ordinary differential equationdefined on some interval [0, T] with T > 0. If q(t) : [0, T] -?R is absolutely continuous and satisfiesandthen q(t) < r(t), t [0,T].Theorem 4 Let w(t, z) E C(R+ x R, R) and let (t) be the minimal solution of the ordinary differential equationdefined on some interval [0. T] with T > 0. If q(t) : [0, T] - R is absolutely rontinuous and satisfieswith q(0) 20 = p(0), then q(t) p(t), t e [0,T].Using above two theorems, we can prove global finiteness for the system:Theorem 5 Consider the problem (4) withwhere p TV, 0 e C(R+, R+), d |
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