| In the present paper, we study the following pseudo-parabolic equations via the critical point theory, variational methods and Nehari manifold ut - a?ut - ?u=f?u?, x €?,t>0, where ? ???RN is an open set. When a=0, the equation mentioned above is a classical heat equation. In the case that a=1, as a pseudo-parabolic equation, it arises in many interesting physical phenomena, for examples, the seepage of homogeneous fluids through a fissured rock and the unidirectional propagation of nonlinear dispersive long waves and so forth.This paper consists of six chapters.In Chapter 1, some research background, the research advance of the related work on pseudo-parabolic equations are given. Moreover, the main results obtained in this thesis are listed.In Chapter 2, using the perturbed potential wells methods, we consider the following initial boundary value problem of pseudo-parabolic equations with a general nonlinearity where ? ??? RN is a smooth bounded domain and f satisfies the following conditions:?f1?f? C1?R,R? and there exist some C>0 and p ??2,2*? such that |f'?u?|?C(1+|u|p-2),u?R, where 2*=2N/?N-2? for N?3 and 2*=? for N=1,2;?f2?f?u?u> 0 for all u?R\{0} and there exists some ?> 1 such that f?u?/|u|? is increasing on ?-?,0? and ?0, ??.Firstly, we investigate the local existence and uniqueness of solutions to the problem mentioned above. The tools used by most mathematicians to do this are the Galerkin method and semigroup theory. If one adopts the Galerkin method, then he needs not only to make some complicated prior estimates, but also prove some convergency. However, during this process, one can only achieve the existence of solutions which have no specifical formulas. Compared with this method, the semigroup theory may provide a represented solution, but a large space is needed to verify that the related operator can generate a semigroup. In Section 2.2, via the elliptic regularity theory, Sobolev embedding and theory about abstract ordinary differential equations in Banach space, we obtain a new representation of solutions to the problem ?2.1.1? which does not depend on any semigroup. At the same time, we also achieve two important identities.Theorem 2.1.4. Assume that ?f1? holds and u0? H01???. Then the problem ?2.1.1? admits a unique solution u? C1?[0,T?,H01???) and u can be represented in the following integral forms or where T is its maximal existence time. Furthermore, if T<?. then Moreover, the following two identities hold and where ||u|12= ??[|?u|2+u2],u? H01???, J, I are the energy and Nehari functionals of the stationary equation of the problem ?2.1.1? respectively.In Section 2.3, we consider the global existence and blowup of solutions to the problem ?2.1.1? and prove that the following theorem holds.Theorem 2.1.5. Suppose that ?f1? and ?f2? hold, u0? H01??? with J?u0??d. and u= u?x,t;u0? is the solution to the problem ?2.1.1? whose maximal existence time is T.??? If I?u0?> 0. then T=?. and for any t0 ? R+ with t0+[d-J{u0)]>0, there exists ?=??to?> 0 such that??? If I?u0?< 0, then T<? and u blows up at T. that is,??? If I?u0? = 0, J?u0?=d. then u0 is an unstable stationary solution to the problem ?2.1.1?.Theorem 2.1.5 extends the conclusions derived for the pure power source by Xu and Su in [78] to a more general case. To achieve this, we use a family of perturbed functionals I?, manifolds N? and infimums d???:=inf N? J. Via the implicit function theorem, variational methods and Sobolev embedding, we prove that for all ?? ?0,??+1?/2?, d??? can be attained, which has simplified the proof about the monotonicity and continuity of d?-?.Subsequently, we establish the boundedness and convergency of global solutions to the problem ?2.1.1? using Theorem 2.1.5.Theorem 2.1.7. Assume that ?f1? and ?f2? hold, u0 ?H01??? and u=u?x,t,u0? is a global solution to the problem ?2.1.1?. Then u ? L??R+,H01????. Furthermore, there exist {tn}, c?0 and u* ?Kc satisfying that tn?? and where Kc={u?H01???:J1?u?= 0,J?u?= c}. Particularly, if u* is an isolated critical point of J. thenIn view of the conclusions of Theorem 2.1.5, we propose and solve three questions in Chapter 3. Moreover, we establish some new sufficient conditions to guarantee the blowup of solutions to pseudo-parabolic equations.The first natural question is whether I?u0?< 0 still makes the solution to the problem ?2.1.1? blow up at a finite time without the limit that J?u0?? d. The answer is proved to be positive; for a special f in Section 3.1. Furthermore, combining the flow decided by the solution u and the Nehari manifold, we also achieve a sufficient and necessary condition to guarantee the blowup of solutions to the problem ?3.1.1?. Actually, let N-={u ? H01???:I{u)< 0}. Then the whole space H01??? can be divided to be H01{?)= S+ ? S-where S+={u0 ? H01???:u?t???? N.,t ? [0, T)}, S-={u0 ? H01???:there exists some t0 ? [0,T) such that u?t0? ? N-}. Here, the character S means "source". Our main result in Section 3.1 is the following theorem.Theorem 3.1.2. ??? Assume that u0 ? H01???,u= u?x,t;u0? is the solution to the problem ?3.1.1? whose maximal existence time is T. Then T< oo if and only if u0 ? S-In other words, T< oo if and only if there exists some t0 ? G [0, T) such that I?u?to??< 0.??? 0 is the unique stable steady solution to the problem ?3.1.1?.Different from the previous literature, ??? of Theorem 3.1.2 gives a criteria for the blowup of weak solutions to the problem ?3.1.1?. which is not really related to the initial data but depends on the corresponding trajectory of the solution. Moreover, we also obtain that S- is invariant under the flow decided by the solution u. In this sense, the blowup of a solution can be linked with some invariant sets related to itself. This is unusual and interesting. So far the main results of this section have been accepted by ?Proceedings of the Royal Society of Edinburgh:Section A Mathematics?, see [83].Because Chen and Tian have proved in [15] that the answer to the first question is negative in the case that f?u?=u log|u|, then we can not expect that for all f. I?u0?< 0 is a sufficient condition making the solution blow up at a finite time. Therefore, the second question arises:whether there exists a suitable functional I satisfying that{v?H01???: I?v?<0,J?v??d}??? 0 and I?u0?< 0 makes the solution blow up at a finite time. Section 3.2 is devoted to finding such a functional. Actually, under the assumptions ?f1? and ?f2?, we prove that the answer is positive and establish a new sufficient condition which makes the solution blow up at a finite time. Define where ?=?+1,v= ?1/?1+?1?, Then we have the following conclusion.Theorem 3.2.1. Suppose that ?f1? and ?f2? hold, u0 ?H01 ???\{0} and u= u{x, t; u0) is a solution to the problem ?2.1.1?. If I?u0??0, then T<? and u blows up at T.Both the two questions proposed above are concerned with the pseudo-parabolic equations ?a=1?. The third question is whether the conclusions derived in Sections 3.1 and 3.2 still hold for the parabolic equation ?a=0?. In Section 3.3, we consider an initial boundary value problem for a classical heat equation. Firstly, we illustrate that Theorem 3.1.2 does not hold for the problem ?3.3.1?. Secondly, motivated by the methods used in Section 3.2, we achieve a new sufficient condition to guarantee the blowup of solutions to the problem ?3.3.1? and obtain a larger blowup set. In fact, we define a functional J* and the related manifold N* as follows where ?1 denotes the first eigenvalue of the operator -? in H01??? with the homogeneous Dirichlet boundary condition. Furthermore, let Here V={u? H01???:J?u?< d,I{u)< 0}. Subsequently, we establish a new sufficient condition which makes the solution blow up at a finite time.Theorem 3.3.3. If u0 ? B. then the solution u to the problem ?3.3.1? blows up at its finite maximal existence time. Particularly, if J*?u0?? 0. then the solution blows up at its finite maximal existence time.Owing to [26, Theorem 9] or [64, Theroem 19.5, p.118], V is a blowup set for the problem ?3.3.1?. In Section 3.3, B is proved to be strictly larger than the set V. Hence, our result has improved the previous conclusions about the blowup of solutions to the problem ?3.3.1?. So far, the main results of Section 3.3 have been published in ?Applicable Analysis: An International Journal?, see [82].In Chapter 4, we investigate a class of pseudo-parabolic equations in R2 with an exponential type source. Assume that f satisfies the following condition and ?f2?:?f'1? f ? C1?R,R? and possesses a subcritical exponential growth, that is. for each ?> 0, there exists a positive constant C? such thatAs a consequence of ?f'1?. f may be an exponential function which leads to that f can not be controlled by a power function at infinity, thereby the condition ?f1? does not hold. Hence, we must reprove the conclusions obtained in Theorems 2.1.4,2.1.5.2.1.7 and 3.2.1 also hold for the problem ?4.1.1?.Theorem 4.1.3. Assume that ?f'1? holds and u0 ? H01???. Then the problem ?4.1.1? admits a unique weak solution u ?C1 ?[0,T?, H01???), where T is its maximal existence time. Particularly, if T< ?, then Moreover, two identities similar to ?2.1.11? and ?2.1.12? still hold, and u can also be represented in an integral form independent of any semigroup.Theorem 4.1.4. Assume that ?f'1? and ?f2? hold, u0 ? H10??? with J?u0??d, and u=u?x,t;uo? is the solution to the problem ?4.1.1? whose maximal existence time is T.??? If I?u0?> 0. then T=? and for any t0 ? R+ with t0+[d-J?u0?}> 0. there exists ?=??to?> 0 such that??? If I?u0?< 0. then T<? and u blows up at T, that is??? If I?u0?= 0, J ?u0?= d, then u0 is an unstable stationary solution to the problem ?4.1.1?.Furthermore, we can also obtain a sufficient condition for the blowup of solutions to the problem ?4.1.1?.Theorem 4.1.5. Assume that?f'1? and ?f2? hold, u0 ?H01???\{0}, and u=u?x, t; u0? is a solution of the problem ?4.1.1? whose maximal existence time is T. If I*?uo??0, then the solution blows up at its finite maximal existence time, where I*?u?=||u||12-???F?u?,u?H01{?).To our best knowledge, there are few literature concerned with the pseudo-parabolic equations with an exponential source. Chapter 4 has been published in ?Communications on Pure and Applied Analysis?, see [85].Chapter 5 is concerned with a class pseudo-parabolic equations with a nonlocal term where ? is a smooth bounded domain in R3, p ??2,6?, a?0, b=±1, ?u is a nonlocal term which is defined by Here, G is the Green function corresponding to ? given by [23, §2.2.4?a?, p.34], that is, where for any given x ??,?x is a correction function satisfying the following boundary problemWe consider the global existence and blowup of solutions to the problem ?5.1.1? by the perturbed potential wells method. Note that owing to the presence of ?u, the solution depends on its range in the whole ?. Then we can not apply the conclusions obtained in Sections 2.1 and 3.2 to the problem ?5.1.1? directly. Therefore, we must verify that Theorems 2.1.4,2.1.5,2.1.7 and 3.2.1 also hold for the problem ?5.1.1?. First, in Section 5.2, we obtain the local existence and uniqueness of solutions to the problem ?5.1.1?Theorem 5.1.2. Let u0 ? H01???, b= ±1 and p ? [2,6]. Then the problem ?5.1.1? has a unique solution u ? C1?[0, T?,H01???) whose maximal existence time is T. Moreover, u can be represented in the following integral forms Furthermore, if T<?, then limSecond, in Section 5.3, we prove an important theorem about the global existence and blowup of solutions to the problem ?5.1.1?.Theorem 5.1.3. Suppose that p satisfies that p ? ?4,6? for b=-1 and p E ?2,6? for b=1, u0?H01 ???, J?u0?? d and u=u?x,t;u0? is the solution to the problem ?5.1.1? whose maximal existence time is T.??? If I?u0?> 0. then T=?, and for any t0?R+ with t0+[d-J?u0?]> 0. there exists ?=??10?> 0 such that??? If I?u0?< 0. then T< oo and u blows up at T, that is,??? If I?u0?= 0, J?u0?= d, then u0 is an unstable stationary solution to the problem ?5.1.1?.At the end of Chapter 5. we establish a criterion to justify the blowup of solutions to the problem ?5.1.1?. Let p=min{2,p/2} andTheorem 5.1.5. Suppose that u0 ? H01??? and p satisfies that p ??4,6? for b= -1 and p E ?2,6? for b=1. If I*?u0?< 0, then the solution u blows up at its finite maximal existence time.Chapter 5 has improved the results in [32], which is concerned with the solutions to the problem ?5.1.1? in the case that a= 0,b=±1 and p ? ?4,6?. So far. some results of Chapter 5 have been published in ?Mathematical Methods in the Applied Science?, see [84],The final chapter is devoted to the steady state to the following nonlocal pseudo-parabolic equations with a critical source in R3 where ? ?R+ and g satisfies the following conditions:?g1? g has a quasi-critical growth, that is,?g2? g?u?u>0,u?R,g?u?/|u|3 is increasing on ?-?,0? and ?0,??, and where G?u?=?0u g?s?ds,u?R.We establish the existence of the stationary solution with the least energy via the perturbation method, that is, we achieve the ground state solution to the Schrodinger-Poisson system ?6.1.2? with a critical growth. Our main result in Chapter 6 is the following theorem.Theorem 6.1.1. Assume that g satisfies the conditions ?g1? and ?g2?. Then there exists a ?0> 0 such that the system ?6.1.2? has a ground state solution for all ?? [0,?0).Different from [81]. during the proof of Theorem 6.1.1, we seek for a nonzero critical point of the functional J\near the mountain pass type critical point of J0 for the case that ?> 0. Since we do not need to verify that the functional J? satisfies the ?PS?C? condition. then we succeed in improving the methods used in [81]. |
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