Weak solutions and blow-up for wave equations of p-laplacian type with supercritical sources

Transcription

1 University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 1 Weak solutions and blow-up for wave equations of p-laplacian type with supercritical sources Pei Pei Earlham College, peipe@earlham.com Mohammad A. Rammaha University of Nebraska-Lincoln, mrammaha1@unl.edu Daniel Toundykov University of Nebraska-Lincoln, dtoundykov@unl.edu Follow this and additional works at: Pei, Pei; Rammaha, Mohammad A.; and Toundykov, Daniel, "Weak solutions and blow-up for wave equations of p-laplacian type with supercritical sources" 1). Faculty Publications, Department of Mathematics This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.

2 JOURNAL OF MATHEMATICAL PHYSICS 6, 813 1) Weak solutions and blow-up for wave equations of p-laplacian type with supercritical sources Pei Pei, 1,a) Mohammad A. Rammaha,,b) and Daniel Toundykov,c) 1 Department of Mathematics, Earlham College, Richmond, Indiana , USA Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska , USA Received 31 January 1; accepted 1 July 1; published online 11 August 1) This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, u tt p u u t = f u), in a bounded domain R 3 and subject to Dirichlét boundary conditions. The operator p, < p < 3, denotes the classical p-laplacian. The nonlinear term f u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W 1, p ) into L ). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy. C 1 AIP Publishing LLC. [ I. INTRODUCTION A. The model The study of quasilinear wave equations with the p-laplacian operator originates from the nonlinear Voigt model for the longitudinal vibrations of a rod made from a viscoelastic material. In particular, it can be shown that the so-called Ludwick materials obey such equations under the effect of an external forcing e.g., see Refs. 37 and 4). This paper addresses the existence of local and global solutions to the following quasilinear problem: u tt p u u t = f u) in,t), u), u t )) = u, u 1 ) W 1, p ) L ), < p < 3, u = on Γ, T), where R 3 is a bounded open domain with a C -boundary Γ. The p-laplacian for p is defined by p u = div u p u). This operator can be extended to a monotone operator between the space W 1, p ) and its dual as follows: p : W 1, p ) W 1, p ), p u, φ p = u p u φdx, p <, where, p denotes the duality pairing between W 1, p ) and W 1, p ), 1 p + 1 p = ) 1.) a) address: peipe@earlham.com b) address: mrammaha1@unl.edu c) address: dtoundykov@unl.edu -488/1/68)/813/3/$3. 6, AIP Publishing LLC

3 813- Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) B. Literature overview and new contributions The semilinear case with the classical Laplace operator when p = ) was studied by Webb, 41 but only under the influence of a conservative restoring ) source f u) that is globally Lipschitz continuous from H 1 ) into L ). In addition, the case p = with a supercritical as defined below) source has been recently treated in Ref. 3, and later in Ref. 33 where the authors provide a complete analysis of that problem including local and global well-posedness, uniqueness, continuous dependence on the initial data, and decay of energy. Other related works include Biazutti 1 investigated global existence and asymptotic behavior of weak solutions to the abstract Cauchy problem with a solution-independent forcing), u tt p u u t + u t ρ sgnu t ) = f t, x). 1.3) In Ref. 16, Gao and Ma studied the global existence and asymptotic behavior of solutions to such equations under fractional-laplacian Kelvin-Voigt damping, u tt p u + ) α u t + gu) = f t, x), 1.4) where α,1]. The source-feedback map g is subject to the growth condition gx,u) a u σ 1 + b, for 1 < σ < pn n p. The first existence theorem in Ref. 16 deals with the case σ < p which is subcritical in our formulation below. And the supercritical scenario was only addressed under a smallness assumption on the initial data Ref. 16,.6) and.7) p. 3. Ma and Soriano in Ref. 8 considered the equation u tt n u u t + gu) = f t, x), where gu) is a dissipative source term, gs)s with the growth bound of the form e s n/n 1) with R n, n. Remark 1.1. The authors of Refs. 1 and 16 cited the above considered wave equations with the pseudo-laplacian operator p u = j x j u p u x j x j ) which is slightly different from 1.). In this paper, we prove well-posedness results for a more general range of nonlinear sources. We focus on the more interesting situation of dimension n = 3 and exponent < p < 3. For p =, the principal part becomes linear, whereas for p 3, the topology of W 1, p ) permits consideration of source terms with arbitrary large exponent, in accordance with the classical Sobolev embedding results in 3D. The source f u) considered in 1.1) need not be dissipative and is allowed to be of a supercritical order. The criticality classification is chosen with respect to the associated Sobolev embeddings. In particular, we assume that f C 1 R) with the following polynomial growth rate at infinity: f s) c s r for all s 1. We follow the terminology introduced in Refs. 1 and 13: In view of the 3D embedding W 1, p ) L 3p 3 p ), with p < 3, the source, if regarded as a Nemytski operator f : W 1, p ) L ), is locally Lipschitz for the values 1 r 3p When the exponent r satisfies 1 r < 3p 3 p). r = If the exponent r satisfies 3p 3 p) 3p 3 p) < r 4p 3 3 p, 3 p)., we call the source subcritical, and critical if the source will be called supercritical, and in this case f is no longer locally Lipschitz continuous from W 1, p ) into L ). However, for this range of exponents, the associated potential energy functional

4 813-3 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) 1 p u p p Fut))dx is well-defined on the finite energy space u,u t ) W 1, p ) L ); here, F denotes a primitive of f. When 4p 3 3 p < r < 3p 3 p, the source is referred to by a somewhat lengthy, but now frequently used term, super-supercritical. In this scenario, the potential energy may not be defined in the finite energy space and the problem itself is no longer well-posed within the framework of potential well theory see Refs. 4, 6, 3, 39, and 4). Remark 1.. It is worth noting here that when the damping term u t is absent, a source term of the form u r 1 u, with r > r p for some r p > 1 dependent on p, should drive the solution of 1.1) to a blow-up in finite time. In such a scenario, by appealing to a variety of methods going back to the works of Glassey, 18 Levine, and others), one can show that a large class of solutions to the problem develop a singularity with respect to the natural energy topology in finite time. On the other hand, if the source f u) is removed from the equation, then it should possess global solutions cf. Refs.,, 7, and ). However, when both damping and source are present, the analysis of their interaction and their influence on the global behavior of solutions becomes more difficult and has served as a motivation for this study see also other related results 8,17,6,31,34,3 and the references therein). Results on existence of solutions to the above problem have a non-trivial history. For secondorder problems, the quasi-nonlinearity of the p-laplacian makes the system non-monotonic, which prevents a direct application of nonlinear semigroup techniques. So the process of Galerkin approximation will be invoked similarly to Refs. 1, 16, and 8. We adapt the strategy in Refs. 11, 14, 19 1, 8, 33, and 3 and show that 1.1) has local weak solutions even in the presence of supercritical sources. The details of the proof, however, are non-trivial when it comes to identifying the weak limits associated to the p-laplacian nonlinearities. Despite this solution strategy being used previously for more restrictive sources) in this context, we believe that this is the first manuscript supplying comprehensive details of such an existence argument. Our detailed approach also highlights the crucial difficulty that would arise if the Kelvin-Voigt damping was replaced with an m-laplacian term m u t, m >. In that case, the simultaneous identification procedure for the two weak limits, one for the p-laplacian of u and the other for the m-laplacian of u t even if m = p) cannot be carried out by the same methods. It had been assumed in some previous works, e.g., Ref. 9 which attempts to rely on Refs. 9 and 1 that deal with a single p-laplace operator in the equation), that the Galerkin approach might trivially extend to the m-p model. That is not the case and rigorous analysis of well-posedness for p-laplacian/m-laplacian with m, p > ) second-order equation is presently missing from the literature, remaining a challenging open problem. In addition to the existence theorem with supercritical sources, the concluding result of this article also verifies that solutions blow up in finite time provided the initial total energy is negative and the source dominates the damping term in an appropriate sense. C. Strategies and technical difficulties Here, we outline the paper highlighting some of the technical steps in the arguments. Section I D summarizes the standing assumptions on the nonlinear source term f and presents the main results of the paper. In addition, the auxiliary result of Lemma 1.1 is nontrivial and plays an important role in the proof of the energy inequality see Section II E) and in the verification of the existence result for more general sources see Sections IV). Sections II IV focus on proving the existence of the local solutions. The strategy used to verify the solvability of 1.1) can be summarized as follows:

5 813-4 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) Step 1: Step : Step 3: For a source that corresponds to a globally Lipschitz Nemytski operator from W 1, p ) into L ), we can construct a local solution by using Galerkin approximations. The particular Galerkin approximations used here are similar to those in Refs. 1, 16, and 8, but this manuscript supplies additional crucial details which cannot be omitted. Extend the existence result in Step 1 to confirm local existence of solutions in the case of sources that are locally Lipschitz from W 1, p ) into L ) by using a standard truncation argument e.g., see Refs. 1 and 3). It is essential to demonstrate that the local existence time T does not depend on the local) Lipschitz constant of f, regarded as a map from a bounded subset of W 1, p ) into L ), but rather depends on the corresponding constant of the source f as a mapping from a subset of W 1, p ) into L 6/ ) again, the exponents are for 3D setting). Construct approximations of the original source that obey the requirements of Step by using smooth cut-off functions introduced in Ref. 3. Finally, pass to the limit in the weak variational form for Galerkin approximations to conclude the existence of a local weak solution to the original problem. An important ingredient in this argument is the energy inequality t Et) + u t s) ds E), where Et) denotes the total energy of the system Et) 1 u tt) + 1 p ut) p p and Fu) = u f s)ds. Fut))dx, Section V proves global existence of solutions as claimed in Theorem 1.4 for the case when the exponent of the damping dominates that of the source. Here, the energy inequality also plays a crucial role as well as the fact that t f uτ))u tτ)dxdτ = Fut))dx Fu))dx, which is non-trivial due to the lack of regularity. Finally, Theorem 1.6 presented in Section VI verifies that solutions blow up in finite time provided the initial total energy is negative and the exponent of the source dominates that of the damping term. Due to the lack of regularity, we have to separately verify the product rule for derivatives Proposition A.1 in the Appendix) in general functional spaces which is one of the technical challenges addressed in Section VI. D. Preliminaries and main results We begin by introducing some basic notation that will be used in the subsequent discussion. Define the usual Lebesgue norms and the L -inner-product u r = u L r ) and u,v) = u,v) L ). The duality pairing between the space W 1, p ) and its dual W 1, p ) will be denoted using the form, p. According to Poincaré s inequality, the standard norm u 1, p W is equivalent to the ) norm u p on W 1, p ). Henceforth, we put u 1, p = u W ) p. The following Sobolev embedding theorem in 3D will be invoked frequently: W 1, p ) L 3p 3 p ), for < p < 3. 1.) Throughout the paper, we assume the validity of the following assumption. Assumption 1.1. Assume that the exponent of the p-laplacian belongs to the range < p < 3, the source feedback function f C 1 R) satisfies for s 1

6 813- Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) for some constants c, c 1 > where f s) c s r, f s) c 1 s r 1 1 r < p 3 p), 1.6) the initial data reside in the following function spaces: u W 1, p ) and u 1 L ). The assumption on source term is very general and f need not be locally Lipschitz continuous form W 1, p ) into L ). However, one can still take advantage of the following lemma: Lemma 1.1. Under Assumption 1.1, then for all sufficiently small ϵ, δ, 1/) satisfying 1.8) and 1.17) below, respectively, we have f : W 1 ϵ, p ) L 6 ) is locally Lipschitz continuous. f : W 1, p ) L 6 1+δ) ) is locally Lipschitz continuous. Proof. Let us identify ϵ > such that for a given R > and all u, v W 1 ϵ, p ) with u W 1 ϵ, p ), v W 1 ϵ, p ) R, the following inequality holds: f u) f v) 6/ C R u v W 1 ϵ, p ), 1.7) where the constant C R > is independent of u and v. First note that by the restriction on r in 1.6), we can choose ϵ, 1 ) such that This inequality readily implies < ϵ < p 3 p) r p 3 p r 1 r < = r 3 p p. 1.8) p pϵ + 3 p). 1.9) It follows from the mean value theorem and Assumption 1.1 that f u) f v) 6/ 6/ = f u) f v) 6 dx = f ξ u,v )u v) 6 dx C u v 6 6 u r 1) + v 6 r 1) + 1 dx. 1.1) 3p Having the embedding W 1 ϵ, p ) L pϵ+3 p ) in mind for instance, see Ref. 1), we employ 3p Hölder s inequality in 1.1) with the conjugate exponents: α = and 6 pϵ+3 p) α 3p = to 3p 6 pϵ+3 p) obtain ) f u) f v) 6/ 6/ C u v 6 3p u 6 r 1) + v 6 r 1) + C 6 r 1)α 6 r 1)α 1, 1.11) pϵ+3 p where C 1 > depends on. It is easy to check that ϵ [, 1/]. According to inequality 1.9), 6 r 1)α = < 6 3p 6 r 1) pϵ + 3 p) 3p 6 p pϵ + 3 p) 1 ) 6 3p pϵ+3 p) > 1 for all p [,3) and all 3p 3p = 3p 6 pϵ + 3 p) pϵ + 3 p. 1.1) Therefore, for any R >, and for all u, v W 1 ϵ, p ) with u W 1 ϵ, p ), v W 1 ϵ, p ) R, inequality 1.11) and the embedding W 1 ϵ, p ) L 3 1 ϵ)p ) imply 3p

7 813-6 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) ) f u) f v) 6/ 6/ C u v 6 u 6 r 1) + v 6 r 1) + C W 1 ϵ, p ) W 1 ϵ, p ) W 1 ϵ, p ) 1 CR 6 r 1) + C 1 ) u v 6 W 1 ϵ, p ), 1.13) which proves 1.7). We now address the second bullet in the statement of the lemma. To this end, we will find δ > such that for any R >, and for all u, v W 1, p ) with u 1, p, v W ) W 1, p R, the source f ) satisfies f u) f v) 61+δ)/ C R u v 1, p, 1.14) W ) where C R > is independent of u and v. From the mean value theorem and Assumption 1.1, we conclude that f u) f v) 61+δ)/ 61+δ)/ = f u) f v) 6 1+δ) dx f ξ u,v )u v) 6 1+δ) dx C u v 6 1+δ) u 6 1+δ)r 1) + v 6 1+δ)r 1) + 1 dx. 1.1) Invoke the embedding W 1, p ) L 3p 3 p ) and Hölder s inequality with exponents α 3p = and 6 1+δ)pϵ+3 p) α 3p = to derive 3p 6 1+δ)pϵ+3 p) f u) f v) 61+δ)/ 61+δ)/ C u v 6 1+δ) 3p [pϵ+3 p] u 6 1+δ)r 1) + v 6 1+δ)r 1) + C 6 1+δ)r 1)α 6 1+δ)r 1)α 1 ), 1.16) where C 1 > depends only on the domain. Inequality 1.9) with < ϵ < 1/ being the same as in 1.8)) implies that there exists some δ,1/) such that < δ)r 1)α < 3p 3 p. 1.17) Therefore, for any R > and for all u, v W 1, p ) with u 1, p, v W ) W 1, p embedding W 1, p ) L 3p 3 p ) along with 1.17) imply f u) f v) 61+δ)/ 61+δ)/ C u v 6 1+δ) which completes the proof of Lemma 1.1. W 1, p ) u 6 1+δ)r 1) W 1, p ) C R 6 1+δ)r 1) + C 1 u v 6 1+δ) ) ) + v 6 1+δ)r 1) + C 1 W 1, p ) R, the Sobolev, 1.18) W 1, p ) In order to state our main results, we begin with a precise definition of a weak solution to 1.1). Definition 1.. A function u is said to be a weak solution of 1.1) on [,T] if u C w [,T], W 1, p )), u t C w [,T], L )) L,T,W 1, )), u),u t)) = u,u 1 ) W 1, p ) L ), and u verifies the identity t u t), φ) u ), φ) + ut) p ut) φdxdτ + t u t t) φdxdτ = for all test functions φ W 1, p ), and for almost everywhere t [,T]. The first result establishes the existence of a weak solution of 1.1). t f ut))φdxdτ, 1.19)

8 813-7 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) Theorem 1.3 Local solutions). Under the validity of Assumption 1.1, problem 1.1) has a local weak solution u defined on [,T] in the sense of Definition 1.) for some T > which depends only on the norms u) 1, p, u W ) t) and p. In addition, u satisfies the following energy inequality: t Et) + u t s) ds E), 1.) for all t [,T], where Et) denotes the total energy of the system Et) 1 u tt) + 1 p ut) p p Fut))dx, 1.1) and Fu) = u f s)ds. Moreover, for all t [,T] the following identity holds: t f uτ))u t τ)dxdτ = Fut))dx Fu))dx. 1.) Remark 1.3. Note that no claims of uniqueness are made here. The next theorem states that the weak solution described by Theorem 1.3 can be extended globally to the time interval [, ) provided the source exponent is at most p/. Theorem 1.4 Global solutions). In addition to Assumption 1.1, assume that r p. Then, the weak solution u furnished by Theorem 1.3 is a global solution and the existence time T can be taken arbitrarily large. In order to state our blow-up result, we need to impose additional assumptions on the source term. Assumption 1.. Assume that the source map is given by f s) = r + 1) s r 1 s, where r 1. In particular, f s) = d ds Fs), where Fs) = s r+1, and s f s) = r + 1)Fs), for all s R. Theorem 1.6 Blow up in finite time). In addition to Assumptions 1.1 and 1., suppose that r > p 1 and the initial total energy is negative, E) <. Then, any weak solution u to 1.1) provided by Theorem 1.3 blows up in some finite time. More precisely, lim sup t T Et) = for some T <, where Et) is the positive energy given by Et) = 1 u tt) + 1 p ut) p p. II. LOCAL SOLUTIONS FOR GLOBALLY LIPSCHITZ SOURCES The first step towards a proof Theorem 1.3 is Proposition.1 which deals with the case of globally Lipschitz sources. Proposition.1. In addition to Assumption 1.1, assume that f : W 1, p ) L ) is globally Lipschitz with a Lipschitz constant L >. Then, system 1.1) has a local weak solution u defined on time-interval [,T], for some T > which depends on the norms of the initial data u) 1, p, W ) u t ). In addition, u satisfies energy inequality 1.) and identity 1.). Remark.1. Due to lack of regularity, energy inequality 1.) is not a trivial corollary of the existence result and will be proved in Section II E. The proof of Proposition.1 will be carried out in five steps outlined in Subsections II A II E.

9 813-8 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) A. Approximate solutions Our strategy here is to use suitable Galerkin approximations. Consider A = as an unbounded operator on L ) with the domain DA) = W 1, ) W, ) which relies on the fact that is bounded of class C ). It is well known that A is positive, self-adjoint, and A 1 is compact. Moreover, A has an infinite sequence of positive eigenvalues λ j ) j=1 and a corresponding sequence of eigenfunctions w j ) j=1 that forms an orthonormal basis for L ). Namely, if u L ), then u = j=1 u j w j, with u j = u, w j and the convergence in the L )-sense; the norm of u is given by u =. j=1 uj Thus, Au = j=1 λ j u j w j and its domain can be equivalently characterized as DA) = u = u j w j L ), Au = λ j u j <. This domain is itself a Hilbert space with the inner product j=1 u,v) DA) = j=1 λ j u jv j and u DA) = Au,.1) j=1 where u = j=1 u j w j and v = j=1 v j w j. Furthermore, the sequence w j ) j=1 forms an orthogonal basis for DA). Hence, for initial data u,u 1 ) W 1, p ) L ), we can find sequences of scalars u N, j ; j = 1,,..., N N=1 and u1 j ) j=1 such that lim N N j=1 u N, j w j = u, in W 1, p ),.) u 1 j w j = u 1, in L )..3) j=1 Let V N denote the linear span of w 1,..., w N ), and P N be the orthogonal projection from L ) to V N. Let u N t) = N j=1 u N, jt)w j be the approximate solution of 1.1) in V N, i.e., u N satisfies the following system of ordinary differential equations: u N t), w j + u N t) p u N t), w j + u N t), w j = P N f u N t))), w j,.4) u N, j ) = u N, j, u N, j ) = u1 j ; for j = 1,..., N..) It is clear that.4) and.) are an initial value problem for a second order N N system of ordinary differential equations with continuous nonlinearities in the unknown functions u N, j and their time derivatives. Therefore, it follows from the Cauchy-Peano Theorem that for every N 1,.4) and.) have a solution u N, j C [,T N ], j = 1,... N, for some T N >. Remark.. The projection P N can be discarded in.4) since PN f u N t))), w j = f u N t)), w j, j = 1,..., N. However, we will keep P N in.4) to handle a technical step in obtaining estimate.13) below. B. A priori estimates Next we show that the existence time T N for system.4) can be replaced by some T > independent of N N. Proposition.. Assume < p < 3 and f : W 1, p ) L ) is globally Lipschitz continuous with a Lipschitz constant L >. Then, there exists a constant T >, independent of N N, such

10 813-9 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) that the sequence of approximate solutions u N ) N to.4) and.) satisfies the following: u N ) N is a bounded sequence in L,T; W 1, p )), u N ) N is a bounded sequence in L,T; L ), u N ) N is a bounded sequence in L,T,W 1, )), u N ) N is a bounded sequence in L,T; W 1, p ))..6) Proof. For any fixed N N, multiply.4) by u N, j t) and sum over j = 1,..., N to obtain 1 d u dt N t) + 1 d p dt u Nt) p p + u N t) = f u N t))u N t)dx..7) Replace t by τ in.7) and integrate over τ [,t] to arrive at t 1 u N t) + 1 p u Nt) p p + u N τ) dτ = 1 u N ) + 1 t p u N) p p + f u N τ))u N τ)dxdτ ) t C u 1, p, u W ) 1 + f u N τ))u N τ)dxdτ..8) By using Hölder s inequality and the fact that the map f : W 1, p ) L ) is globally Lipschitz continuous, the last term on right-hand side of.8) can be estimated as follows: f u N τ))u N τ)dx f un τ)) f ) + f ) u N dx f u N τ)) f ) + f ) u N τ) L u N τ) p + c 1 u N τ),.9) where c 1 = f ). By Poincaré s and Young s inequalities, we have f u N τ))u N τ)dx 1 L λ u Nτ) p + λ u N τ) ) 1 + λ c 1 + λ u N τ) ) CL, λ, f )) u N τ) p + 1 ) + λ L + 1) u N τ),.1) where.9) hold for all λ >. Let λ = 1 L+1 and y Nt) = 1.1) imply that y N t) + 1 t u N τ) dτ C + C u N t) + 1 p u Nt) p p, then.8) and t y N τ)) p dτ,.11) where C = C u 1, p, u W ) 1, L, f ) > and C > is some constant dependent on p, L, and f ). In particular, y N satisfies the inequality y N t) C + C t y N τ)) p dτ..1) By a standard comparison theorem see, for instance, Ref. 4), inequality.1) yields y N t) zt), where zt) = C 1 p + C 1 ) 1 1 p t p is the solution of the Volterra integral equation zt) = C + C t zτ)) p dτ.

11 813-1 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) Let us note that since < p < 3, then zt) is defined for all t. Therefore, we can select T > such that y N t) zt) C T < for all t [,T], where C T is independent of N. Hence, for all N 1, the bound y N t) C T holds on [,T], establishing the first two claims in.6). The third claim follows immediately from.11). Finally, to prove the last statement in.6) let N S = α j w j : α j R, N N. j=1 Since S is dense in W 1, p ), then for any φ W 1, p ) there exists a sequence φ j ) j=1 S such that lim φ j = φ in W 1, p j ). Apply identity.4) with w j = φ j and take the limit j to obtain u N t), φ p = u N p u N, φ u N, φ + P N f u N )), φ u N p 1 p φ p + u N φ + f u N ) φ C u N p 1 p + u N + f u N) ) φ 1, p W ) C u N p 1 p + u N C u N p 1 p + u N + f u N) f ) + f ) ) φ 1, p W ) + L u N p + f ) ) φ 1, p,.13) W ) where we have appealed to the assumption that f : W 1, p ) L ) is Lipschitz. Hence, u N t) W 1, p ) C u Nt) p 1 p + u N t) + L u Nt) p + f ) ), for all t [,T]. Therefore, by using the first three claims in.6) it follows that u N W 1, p ) L,T) with the norm bound independent of N. This step completes the proof. The following corollary is an immediate consequence from Proposition. and standard compactness results see, for instance, Refs. 36 and 38). Corollary.3. Assume the sequence of approximate solutions u N ) N satisfies.6). Then, there exist a function u and a subsequence of u N ) N again reindexed by N), such that where ϵ > is as defined in Lemma 1.1. u N u weakl y in L,T; W 1, p )),.14) u N u weakl y in L,T; L )),.1) u N u strongl y in L,T; W 1 ϵ, p ),.16) u N u strongl y in L,T; L )),.17) u N u weakl y in L,T; W 1, )),.18) C. Passage to the limit By integrating.4) over t [,T], we obtain u N t), w j) u N ), w j) + + t t u N w jdxdτ = t un p u N w j ) dxdτ f u N )w j dxdτ..19) In order to pass to the limit in.19), we shall need several auxiliary results. The following lemma addresses the last term in.19).

12 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) Lemma.4. If sequence u N ) N satisfies.6) and f : W 1, p ) L ) is globally Lipschitz with the Lipschitz constant L >, then there exists a subsequence u N ) N re-indexed again by N) which satisfies f u N ) f u) weakl y in L,T; L ))..) Proof. Since f : W 1, p ) L ) is globally Lipschitz continuous and u N ) N is bounded in L,T; W 1, p )), then for all t [,T] f u N t)) = f u N t)) f ) + f ) f u N t)) f ) + f ) L u N t) 1, p + f ) 1 W ) LCT + f ) 1 <..1) Thus, f u N )) N is a bounded sequence in L,T; L )) and thus, there exists ξ L,T; L )) such that, for a suitable subsequence f u N ) ξ weakly in L,T; L )). However,.16) implies that there is a subsequence still denoted as u N ) N ) such that u N u almost everywhere in [,T]. Consequently, by the continuity of f, one has A standard analysis result implies f u N ) f u) point-wise a.e. on [,T]. f u N ) f u) weakly in L [,T]). Hence, ξ = f u) almost everywhere in [,T], completing the proof of Lemma.4. Remark.3. Lemma.4 implies that t f u N )w j dxdτ = lim N t f u)w j dxdτ, for all j N. The next result addresses the passage to the limit in the term containing the p-laplacian. Indeed, identifying the weak limit of the sequence p u N ) N with p u is a key step in the proof. Lemma.. Let X = L p,t; W 1, p )). If u N ) N and u satisfy.14)-.18), then where X = L p,t; W 1, p )) is the dual of X. p u N p u weakl y in X,.) Proof. Since 1 p + 1 p = 1, we note that t un p p t u N dxds = u N p p C T, for all t [,T]..3) Thus, u N p u N ) N is a bounded sequence in L p,t; L p )) 3 and so, there exists a reindexed) subsequence such that u N p u N ψ weakly in L p,t; L p )) 3,.4) for some ψ L p,t; L p )) 3. Let φ X and, X, X) be the duality pairing between X and X, then from.4), we have, as N, T T p u N, φ X, X) = u N p u N φdxds ψ φdxds. Thus, p u N ) N converges weakly* in X. By a standard theorem see Ref. 43, Theorem 7 p.14 for instance), X is sequentially weakly complete. Hence, there is η X such that p u N η weakly in X..)

13 813-1 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) In order to show that η coincides with p u in X we will use the fact that the operator p : X X is maximal monotone see Ref. 3, for example). It follows from.14),.), and [Ref. 6, Lemma 1.3 p. 49) ] that η = p u in X, provided we demonstrate the inequality In order to establish.6), we first note that p u N η,u N u X, X) lim sup p u N η, u N u X, X)..6) N = p u N,u N X, X) p u N,u X, X) η,u N u X, X). Now recall.14), which implies that u N u weakly in X. Hence, We also note that.) yields Now taking lim sup N on both sides of.7), we obtain.7) lim η,u N u X N, X) =..8) lim pu N,u X N, X) = η,u X, X)..9) lim sup p u N η, u N u X, X) N = lim sup p u N, u N X, X) η,u X, X). N Thus,.6) will follow if we prove T lim sup p u N, u N X, X) = lim sup N N η,u X, X). u N p u N τ) u N τ)dxdt.3).31) We will demonstrate.31) in the following two steps. Step 1. We claim that for almost everywhere t [,T] and for an appropriate subsequence of u N ) N we have lim sup N t u N p u N τ) u N τ)dxdτ t u t),ut)) + u ),u)) + u τ) dτ 1 t ut) u) ) + f uτ))uτ)dxdτ..3) In order to verify.3) fix N N. Then, multiply.4) by u N, j t) and sum over j = 1,..., N, to arrive at u N t),u Nt)) + u N p u N t), u N t)) + u N t), u Nt)) = f u N t)),u N t))..33) Relabel t by τ and integrate identity.33) over τ [, t] to obtain t u N p u N τ) u N τ)dxdτ = u N t),u Nt)) + u N ),u N)) + 1 u Nt) u N ) ) + t t u N τ) dτ f u N τ))u N τ)dxdτ..34) Next, we handle each term in.34) as follows. The term u N t), u Nt)). From.16), we know that there exists a subsequence of u N ) N, still denoted by u N ) N, such that u N u in L,T). Hence, on a subsequence, u N t) ut)

14 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) strongly in L ) for almost everywhere t [,T]. Likewise, it follows from.17) that there exists a subsequence of u N ) N, re-indexed by N, such that u N t) u t) strongly in L ) for almost everywhere t [,T]. Then, from.6),.16), and.17) we conclude that lim N u N t),u Nt)) = u t),ut)) for a.e. t [,T]..3) The term u N ), u N)). From.) and.3), we know that u N ) u) strongly and u N ) u ) strongly in L ). Therefore, The term t u N τ) lim N u N ),u N)) = u ),u))..36) dτ. It follows immediately from.17) that lim N t u N t) dτ = t u t) dτ for all t [,T]..37) The term u N t). Invoke Proposition A. in the Appendix, and results.14),.16). Conclude that there exists a subsequence of u N ) N, still denoted as u N ) N, such that u N t) ut) weakly in H 1) for almost everywhere t [,T]. By the weak lower semicontinuity of the L p norms, we conclude that for almost everywhere t [,T], lim sup u N t) = lim inf u Nt) ut)..38) N N The term u N ). An immediate consequence of.) is that The term t lim u N) = u)..39) N fu Nτ))u N τ)dxdτ. From.) and.16), we conclude that t t f u N τ))u N τ)dxdτ = f uτ))uτ)dxdτ..4) lim N Finally, take lim sup as N on both the sides of.34) and combine.3)-.4). Then,.3) follows. Step. Here, we show that t η, uτ) pdτ is equal to the right-hand side of.3) for almost everywhere t [,T], t η,uτ) p dτ = u t),ut)) + u ),u)) + 1 u t) u) ) + t t u τ) dτ f uτ)),uτ)) dτ..41) In order to establish.41), fix j N, multiply.4) by an arbitrary function θ C 1 [,T], and integrate it from to t to obtain u N t),θt)w j) u N ),θ)w j) + t t We now pass to the limit in.4) term by term. = t p u N,θτ)w j p dτ + u N τ),θ τ)w j ) dτ t u N τ),θτ) w j) dτ f u N τ)),θτ) w j ) dτ..4) It has already been shown that u N t) u t) strongly in L ) for almost everywhere t [,T], and u N ) u ) strongly in L ). Consequently, lim N u N t),θt)w j) = u t),θt)w j ) for a.e. t [,T],.43) lim N u N ),θ)w j) = u ),θ)w j )..44)

15 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) It follows directly from.17) and.18) that t lim N u N τ),θ τ)w j ) dτ = lim N t From.) and.), we obtain t lim N t lim N t u N τ),θτ) w j) dτ = u τ),θ τ)w j ) dτ for all t [,T],.4) t p u N,θτ)w j p dτ = f u N τ)),θτ)w j ) dτ = u τ),θτ) w j ) dτ..46) t t η,θτ)w j p dτ,.47) f uτ)),θτ)w j ) dτ..48) Take the limit as N on both sides of.4) and combine.43).48), to derive t η,θτ)w j p dτ = u t),θt)w j ) + u ),θ)w j ) + + t t u τ),θ τ)w j ) dτ t u τ),θτ) w j ) dτ f uτ)),θτ)w j ) dτ for all j N, a.e. t [,T]..49) Since.49) holds for each j N, we may replace θτ)w j by u N τ) to get t η,u N τ) p dτ = u t),u N t)) + u ),u N )) + + t t u τ),u N τ)) dτ t u τ), u N τ)) dτ f uτ)),u N τ)) dτ for a.e. t [,T]..) Let us note here that.14) implies that u N u weakly in L,T; L )). Whence, t lim N u τ), u N τ)) dτ = t u τ), uτ)) dτ = 1 ut) u) ), for all t [,T]..1) By recalling the already established properties of a subsequence of) u N ) N :.),.14),.16),.17), and.1), we can pass to the limit in.) to verify.41). Now, it follows from.3),.41) and Ref. 6, Lemma 1.3 p. 49) that η = p u in L p,t; W 1, p )) for almost everywhere t [,T], which completes the proof of Lemma.. 1. Verification that the limit is a weak solution Here, we show that u satisfies equality 1.19) in Definition 1. of weak solutions. More precisely, for almost everywhere t [,T] and for all φ W 1, p ), u must satisfy u t), φ) u ), φ) = t t u p u, φ dτ u t, φ) dτ + t f uτ)), φ) dτ..)

16 813-1 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) From the fact that η = p u provided in Lemma.), we know that taking θt) = 1 in.49) gives u t), w j ) u ), w j ) + = t t p u, w j p dτ + t u τ), w j ) dτ f uτ)), w j ) dτ for all j N, and for a.e. t [,T]..3) Since S = N j=1 α jw j : α j R, N N is dense in W 1, p ), then for any φ W 1, p ) there exists a sequence φ n ) n=1 S such that lim φ n = φ in W 1, p n ). Thus, by replacing w j by φ n in.3), we have u t), φ n ) u ), φ n ) + = t t f uτ)), φ n ) dτ. p u, φ n p dτ + t u τ), φ n ) dτ.4) Take the limit as n on both sides of.4) to verify that.) holds for arbitrary φ W 1, p ) and almost everywhere t [,T]. D. Additional regularity in time At this point, recall that the constructed weak solution u of 1.1) satisfies u L,T; W 1, p )), u t L,T; L )) L,T; W 1, )), and u satisfies equality 1.19). We still need to check that u C w [,T],W 1, p )), and u t C w [,T], L )). In order to do so, the following lemma addressing the regularity of u will be needed. Lemma.6. Assume that u L,T; W 1, p )), u L,T; L )) L,T; W 1, )), and u satisfies 1.19). Then, and p u L,T; W 1, p )), where W 1, p ) is the dual space of W 1, p ). u L,T; W 1, p )), u L,T; W 1, p )) Proof. Let, denote the duality pairing between W 1, p ) and W 1, p ) or the duality pairing between W 1, ) and W 1, ). Now, the fact that pu L,T; W 1, p )) is trivial, since u L,T; W 1, p )) and p ut), φ = u p u, φ C ut) p 1 φ ) W 1, p ) C T φ 1, p, W ) Similarly, because u L,T; L )), one can show that W 1, p 1, p for every φ W ) and all t [,T]..) u t), φ = u t), φt)) u φ C u φ 1, p,.6) W ) for every φ W 1, p ) and all t [,T]. It follows from.) and.1) that for every φ W 1, p ), u t), φ = d dt u t), φ = d dt u t), φ) u p u, φ + u, φ) + f u), φ).7) C u W 1, p φ ) W 1, p + ) C u W 1, ) φ W 1, p + LC ) T + f ) ) φ. Estimates.) and.6) imply u L,T; W 1, p )).

17 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) Thus, so far, we have proven u L,T; W 1, p )), u t L,T; L )) L,T; W 1, )), In addition, u verifies equality 1.19) and satisfies the conclusion of Lemma.6. It now follows from standard results Ref. 7, Lemmas 8.1 and 8., pp after possibly a modification on a set of measure zero) that u C w [,T],W 1, p )) and u t C w [,T], L )).8) as required by Definition 1.. E. Proof of the energy inequality We show here that any weak solution u to 1.1) described by Proposition.1 satisfies energy inequality 1.) and equality 1.). Proof. Multiply.4) by u N, j t), sum over j = 1,,..., N, and integrate on [,t], t t u N τ)u N τ)dxdτ + u N p u N u N τ)dxdτ Since u N is regular, then t + d 1 dt u N τ) dτ = t d 1 dt u N τ) ) = ) p u Nτ) p p = d Fu N t))dx = dt f u N τ))u N τ)dxdτ..9) u N τ)u N t)dx,.6) u N p u N u N t)dx,.61) f u N t))u N t)dx..6) Define E N t) = 1 u N τ) + 1 p u Nτ) p p Fu N t))dx, then it follows from.9).6) that E N t) + t u N τ) dτ = E N)..63) Next, we pass to the limit in.63). First, by mean value theorem, we have Fu N t)) Fut))dx f ξ) u N t) ut) dx c u N t) r + ut) r + 1) u N t) ut) dx,.64) where r is as defined in 1.6), and ξ = λu N t) + 1 λ)ut) for some λ,1). All the terms on the right-hand side of.63) can be estimated in the same manner. In particular, we can find 1 < δ, δ < 3p 3 p with 1 δ + 1 δ = 1 such that u N t) ut) u N r dx u N t) ut) δ u N t) r rδ..6) Specifically, we can pick δ = 3p1 ϵ) 3 p and δ = 3p1 ϵ) 4p 3 3pϵ any < ϵ < 4p 3) p 1, we have δ < 6 and rδ < 6 for some small enough ϵ > ; indeed, for p 3 p) = 3p 3 p. Since u N ) N is bounded in L,T; W 1, p )) and the 3D) embedding : W 1, p ) L s ) is compact for all 1 s < 3p 3 p, then for a subsequence, still denoted as u N) N, we have u N u

18 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) strongly in L,T; L δ )). In addition, u N ) N is bounded in L,T; L rδ )). Therefore, it follows from.6) that u N t) ut) u N r dx u N t) ut) δ u N t) r rδ,.66) as N, for almost everywhere t [,T]. Similarly, one can prove that un t) ut) u r + u N t) ut) ) dx,.67) as N, for almost everywhere t [,T]. Combine.64).67) to conclude Fu N t))dx = lim N Fut))dx..68) Also, since u N ) u) strongly in W 1, p ), we have lim Fu N ))dx = Fu))dx..69) N Next, recall.14),.18),.63),.68), and appeal to the weak lower semicontinuity of the L q norms. Then, one has for almost everywhere t [,T], Et) + t u τ) dτ lim inf N EN t) + t u N τ) dτ = lim inf N E N)..7) Since u N ),u N ) ) u),u )) strongly in W 1, p ) L ) then with the help of.69) we have lim E N) = E)..71) N Therefore, energy inequality 1.) follows. Now let us prove another version of energy inequality which will be needed in Section III, t t Et) + u t τ) dτ E) + f uτ))u t τ)dxdτ.7) for all t [,T], where Et) = 1 u tt) + 1 p ut) p p. To verify.7), we need to show equality 1.): for almost everywhere t [,T] t Fut))dx Fu))dx = f uτ))u t τ)dxdτ..73) It follows from.6),.68), and.69) that Fut))dx Fu))dx = lim N Therefore,.73) holds provided we show that t t f u N τ))u N τ)dxdτ = lim N = lim N Fu N t))dx t ) Fu N ))dx f u N τ))u N τ)dxdτ. f uτ))u τ)dxdτ, a.e. t [,T]..74) From Lemma 1.1, we know that f : W 1, p ) L 6 1+δ) ) is locally Lipschitz continuous, and also f : W 1 ϵ, p ) L ) 6 is locally Lipschitz continuous. Then, by Hölder s inequality, we have t f un τ))u N τ) f uτ))u τ) ) dxdτ t f un )u N u ) t dxdτ + f u N ) f u) u τ) dxdτ

19 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) t C f t f u N ) 61+δ) u N W 1, p u N u t 61+δ) dτ + f u N ) f u) 6 u τ) 6 dτ 1+6δ ) + 1) u N u 61+δ) 1+6δ C f,t un L,T ;W 1, p + 1) u )) N u L,T ;L θ )) t dτ + u N u W 1 ϵ, p ) u τ) 6 dτ ) + u N u L,T ;W 1 ϵ, p )) u τ) L,T ;L 6 ))),.7) where θ = 61+δ) 1+6δ < 6. Since u L,T; W 1, )), then the Sobolev embedding W 1, ) L 6 ) gives u L,T; L 6 )). So, by.16) we conclude that lim u N u L N,T ;W 1 ϵ, p )) u τ) L,T ;L 6 )) =..76) Since the embedding W 1, ) Lθ ) is compact and u N ) N is bounded in the space L,T; W 1, p )), then by Aubin s compactness theorem, there exists a subsequence of u N ) N, still denoted as u N ) N, such that u N u strongly in L,T; L θ )). It follows from.6) that lim un N L,T ;W 1, p + 1 u )) N u L,T ;L θ =..77) )) Combine.7),.76), and.77) to get.74). The proof of Proposition.1 is now complete. III. LOCAL SOLUTION FOR LOCALLY LIPSCHITZ SOURCES In this subsection, we relax the conditions on the source term and allow f to be locally Lipschitz from W 1, p ) into L ). Proposition 3.1. In addition to Assumption 1.1, assume that f : W 1, p ) L ) is locally Lipschitz. Then, system 1.1) has a local weak solution u, in the sense of Definition 1., on [,T ] for some T > dependent on initial data u,u 1, f ), and the appropriate local Lipschitz constant of the mapping f : W 1, p ) L 6 ). Moreover, u satisfies energy inequality 1.) and equality 1.). Remark 3.1. By assumption, the mapping f : W 1, p ) L 6 ) is a fortiori locally Lipschitz. However, it is essential to note here that the local existence time T in Proposition 3.1 does not depend on the local Lipschitz constant of f as a map from W 1, p ) to L ). Proof. We use a standard truncation of the sources for instance, Refs. 14 and 1). Let Et) = 1 u tt) + 1 p ut) p p denote the positive energy and put f u), if u p K, ) f K u) = Ku f, if u u p > K, p where K is a positive constant such that K > 4E). With the truncation of the source above, we consider the following K) problem: K) u tt p u u t = f K u) in,t), u),ut ) W 1, p ) L ), u =, on Γ,T). We note here that for each such K, the operators f K : W 1, p ) L ) are globally Lipschitz continuous see Ref. 1). Therefore, by Proposition.1, the K) problem has a local weak solution u K defined on [,T] where T depends on u, u 1, and f K. Since f K is also globally Lipschitz continuous from W 1, p ) L 6 ), there exists a constant L f K) > such that f K u) f K v) 6 L f K) u v) p, for all u,v W 1, p ). 3.1)

20 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) In what follows, we shall for brevity denote u K t) by ut). According to Proposition.1, u K satisfies the following energy inequality: t t Et) + u t τ) dτ E) + f K uτ))u t τ)dxdτ. 3.) We now estimate the terms on the right-hand side of 3.) with the help of Hölder s and Young s inequalities, f K uτ))u t τ)dx C ϵ f K uτ)) f K ) 6 + f K ) 6 + ϵ ut τ) 6 where C f = C ϵ f K ) 6. Thus, C ϵ L f K) uτ) p + C f + ϵ u t τ) 6, 3.3) f K uτ))u t τ)dx C K Eτ) p + C f + ϵc u t τ), 3.4) where C K = C ϵ L f K). It follows from 3.), 3.4), and the fact p 1 that Et) + t u t τ) dτ t t E) + C K Eτ)dτ + C K, f T + ϵc u t τ) dτ, 3.) for all t T, where T > will be chosen below and C K, f = C K + C f. By choosing ϵ > sufficiently small, we obtain t t Et) + c ϵ u t τ) dτ E) + C K, f T + C K Eτ)dτ, 3.6) for all t T. Gronwall s inequality gives Et) E) + C K, f T )e C K t, for all t [,T ]. 3.7) Now we recall that K > 4E) and choose K 4E) 1 T = min,t, ln. 3.8) C K We then have 4C K, f Et) E) + C K, f T )e C K t K 4 ec K t K, for all t [,T ]. 3.9) Therefore, the definition of Et), 3.9), and the condition < p < 3 imply that ut) p K, t [,T ]. Thus, f K ut)) = f ut)) on the interval [,T ], and so the considered solution of K) problem 3.1) is, in fact, a solution u of original problem 1.1) on [,T ]. The fact that u satisfies energy inequality 1.) and equality 1.) follows trivially from Proposition.1, completing the proof. IV. LOCAL SOLUTION FOR MORE GENERAL SOURCES Now, we further relax the conditions on the source. Specifically, we will allow for f C 1 R) with the following growth restrictions: f s) c s r, f s) c 1 s r 1, for s 1, for some constants c, c 1 >. Throughout this section, the exponent of the source satisfies 1 r < p 3 p).

21 813- Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) Before completing the proof of Theorem 1.3, additional preparation will be needed. Recall Lemma 1.1 which established, under Assumption 1.1, that f : W 1 ϵ, p ) L 6 ) is locally Lipschitz for some ϵ >. However, since f is not in general locally Lipschitz from W 1, p ) into L ), we shall construct Lipschitz approximations of f. Consider a sequence of smooth cut-off functions η n, as introduced in Ref. 3. More precisely, we choose a sequence η n C R) that satisfies η n 1, for some constant C independent of n). Define η n u) C n, and η n u) = 1, u n, η n u) =, u > n, 4.1) f n u) f u)η n u). 4.) Lemma 4.1. Under Assumption 1.1, for each n N, the function f n has the following properties: f n : W 1, p ) L ) is globally Lipschitz continuous with Lipschitz constant possibly dependent on n. f n : W 1 ϵ, p ) L 6 ) is locally Lipschitz continuous. Furthermore, on any bounded set the local Lipschitz constant does not depend n. Here, the parameter ϵ is as defined in Lemma 1.1. Proof. The proof is very similar to Ref. 3, Lemma.3, and thus it is omitted. A. Approximate solutions and passage to the limit In order to prove the existence statement in Theorem 1.3, we approximate the original problem 1.1) by using the cut-off functions η n introduced in 4.1). In particular, consider the nth problem given by utt n p u n ut n = f n u n ) in,t), u n ),ut n ) = u n,,u n,1 ) W 1, p ) L ), u n =, on Γ,T), where f n = f η n as defined in 4.), and u n,,u n,1 ) u,u 1 ) in W 1, p ) L ), as n, with E n ) E) + 1 for all n N. Recall that E n ) = 1 un,1 + 1 p p un, p, and E) = 1 u p u p p. We would like to apply Proposition 3.1 to nth problem 4.3). In order to do so, we recall the second statement in Lemma 4.1 which guarantees that on any bounded set the local Lipschitz constants of f n : W 1, p ) L ) 6 are independent of n. Hence, by the proof of Proposition 3.1, the local existence time depends on the choice K > 4E n ). Moreover, by choosing K > 4E) + 1) in the proof of Proposition 3.1, we have one K that properly bounds the norms of the initial data for each n N. Therefore, it follows from Proposition 3.1 that for each n N, nth problem 4.3) has a local weak solution u n on [,T] for some T > independent of n), and u n satisfies the energy inequality t E n t) + u n t τ) t dτ E n) + f n u n τ))ut n τ)dxdτ 4.4) for all t [,T]. By the same analysis used to obtain 3.6) and 3.7), we conclude that there exists C T > independent of n such that E n t) + t 4.3) u n t τ) dτ C T for all t [,T]. 4.) Now, by employing standard compactness theorem see, for instance, Ref. 38), there exist function u and subsequence of u n ) n, which we still denote by u n ) n, such that

22 813-1 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) u n ) n is a bounded sequence in L,T; W 1, p ) ), 4.6) u n t ) n is a bounded sequence in L,T; L ), 4.7) u n t ) n is a bounded sequence in L,T; W 1, )). 4.8) Moreover, for each n N, the function u n satisfies t ut n t), φ) ut n ), φ) + + t for all φ W 1, p ). We know that u n t φdxdτ = u n p u n φ ) dxdτ t u n tt t), φ = d dt un t t), φ = d dt un t t), φ) f u n )φdxdτ, 4.9) u n p u n, φ) + u n t, φ) + f u n ), φ) u n p φ p + ut n φ + f u n ) 6 φ 6 4.1) u n p + c ut n ) φ p + f u n ) f ) 6 + f ) 6) φ 6 u n p + c ut n ) φ p + C f u n 1, p + f ) W ) 6 ) φ 6, wherein we have invoked the second statement of Lemma 4.1 for the last inequality. Since Lemma 4.1 assures us that the local Lipschitz constant of f n : W 1, p ) L 6 ) is independent of n only on the norm of u n which is, in turn, uniformly bounded in n), we conclude that From Corollary.3, we know utt) n n is a bounded sequence in L 1, p,t; W ) ). 4.11) u n u weakly in L,T; W 1, p )), 4.1) ut n u t weakly in L,T; L )), 4.13) u n u strongly in L,T; W 1 ϵ, p ), 4.14) ut n u t strongly in L,T; L )), 4.1) ut n u t weakly in L,T; W 1, )), 4.16) where ϵ is as defined as in 1.8). In order to pass to the limit in 4.9) and show that u actually solves 1.1), we need the following lemma. Lemma 4.. If u N ) N and u satisfy 4.14), then f N u N ) f u) weakly in Y, where Y = L 6,T,W 1 ϵ, p )) and Y is its dual space, and where ϵ > is sufficiently small as defined in Lemma 1.1. Proof. We first pick a proper ϵ >. Recall that Lemma 1.1 requires ϵ to obey < ϵ < r 3 p p. Here, we choose < ϵ < min r 3 p p, 3p ) p 3p This choice of ϵ implies that 6 < 3 1 ϵ)p. It follows from the embedding W 1 ϵ, p ) L that 3p 3 1 ϵ)p ) φ 6 C φ W 1 ϵ, p ), for all φ W 1 ϵ, p ). 4.18) By Hölder s inequality and by 4.18), we have

23 813- Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) t f N u N τ)) f uτ)) φdxdτ t T C f N u N τ)) f N uτ)) 6 φ W 1 ϵ, p ) dτ I f N u N τ)) f uτ)) 6 φ 6 dτ T + C f N uτ)) f uτ)) 6 φ W 1 ϵ, p ) dτ, 4.19) I I for all φ Y and all t [,T]. By the second part of the statement of Lemma 4.1 and by convergence result 4.14), we have T I C R u N u W 1 ϵ, p ) φ W 1 ϵ, p ) dτ C R u N u L,T,W 1 ϵ, p )) φ L 6,T,W 1 ϵ, p )), as N. 4.) From Hölder s inequality, we obtain T II C f N uτ)) f uτ)) dτ φ L 6,T,W 1 ϵ, p )) = C f N uτ)) f uτ)) 6 φ L L 6,T,W 1 ϵ, p )). 4.1) [,T ]) Since η N 1 from below as N, then f N u) f u) almost everywhere on [,T]. In addition, by 4.17) we have 6r < 3p 3 1 ϵ)p see 1.9) in the proof of Lemma 1.1). Hence, it follows from the regularity u L,T; W 1 ϵ, p )) and the 3D embedding: W 1 ϵ, p ) L 3 1 ϵ)p ) that f N u) f u) 6 f u) 6 C u 6r + 1) L 1 [,T]). Now Lebesgue dominated convergence theorem gives us T f N uτ)) f uτ)) 6/ 6/ dτ as N. 4.) Finally, combine 4.19) 4.) to complete the proof of Lemma 4.. Now using the same argument as in Sections II C II E, the proof of Theorem 1.3 can be easily completed. 3p V. GLOBAL EXISTENCE This section is devoted to the existence of global solutions, as stated in Theorem 1.4. Here, we appeal to a standard continuation procedure in order to conclude that either the weak solution u is global or there exists < T < such that where Et) is the positive energy defined by lim sup Et) =,.1) t T Et) 1 u tt) + 1 p ut) p p..) We note here that in view of the 3D embedding W 1, p ) L 3p 3 p ) and r + 1 < the energy Et) is well defined for all t [,T]. We aim at proving that.1) cannot happen under the assumption of Theorem 1.4. p 3 p) + 1 < 3p 3 p, Proposition.1. Let u be a weak solution of 1.1) on [,T] as furnished by Theorem 1.3. Then, we have the following.

24 813-3 Pei, Rammaha, and Toundykov J. Math. Phys. 6, 813 1) If r p/, then for all t [,T], u satisfies Et) + t u t τ) dτ CT, E)),.3) where T > is being arbitrary. If r > p/, then the bound in.3) holds for t < T, for some T > dependent on E) and T. Proof. First, revisit energy inequality.7), Et) + t u t τ) dτ E) + t f uτ))u t τ)dxdτ..4) Now we proceed to estimate the last term on the right-hand side of.4). Recall the assumption f s) c s r for s 1 and define Q t,t), Q t {x, τ) Q t : ux, τ) 1}, Q t {x, τ) Q t : ux, τ) > 1}. By Young s inequality t f uτ)) u t τ)dxdτ C C Q t t Q t u t τ) dxdτ + C f uτ))u t τ) dxdτ + Q t Eτ)dτ + C Q T + C Q t uτ) r u t τ) dxdτ t f uτ))u t τ) dxdτ uτ) r u t τ) dxdτ,.) for all t [,T], where Q T denotes the lebesgue measure of Q T. Thus, it follows from.4) and.) that t t Et) + u t τ) dτ E) + C Q T + C Eτ)dτ t + C u t τ) uτ) r dxdτ,.6) for t [,T]. By Hölder s inequality along with Sobolev embedding result 1.) with 6r < 3p 3 p ), we obtain t t uτ) r u t τ) dxdτ u t τ) 6 uτ) r 6r dτ.7) Again by Young s inequality, we have t uτ) r u t τ) dxdτ ϵ Case 1. If r p/, then t uτ) r u t τ) dxdτ ϵ ϵ t t t C 1 u t τ) uτ) r p dτ..8) t It now follows from.6) and.1) that for t [,T], Et) + t + C ϵ u t τ) dτ + C ϵ t uτ) r p dτ..9) t u t τ) dτ + C ϵ,t uτ) p p dτ + C ϵ,t t u t τ) dτ + C ϵ,t Eτ)dτ + C ϵ,t..1) u t τ) dτ E) + C Q T + C t t Eτ)dτ t u t τ) dτ + C ϵ,t Eτ)dτ + C ϵ,t..11)