# Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

Authors: Zhigang Pan; Hong Luo; Tian Ma
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We consider the global existence of strong solutionu, corresponding to a class of fully nonlinear wave equations with strongly damped terms utt-kΔut=f(x,Δu)+g(x,u,Du,D2u) in a bounded and smooth domain Ω in Rn, where f(x,Δu) is a given monotone in Δu nonlinearity satisfying some dissipativity and growth restrictions and g(x,u,Du,D2u) is in a sense subordinated to f(x,Δu). By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution u∈Lloc∞((0,∞),W2,p(Ω)∩W01,p(Ω)).


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## Body

## 1. Introduction

We are concerned with the following mixed problem for a class of fully nonlinear wave equations with strongly damped terms in a bounded andC∞ domain Ω⊂Rn:
(1.1)utt-kΔut=f(x,Δu)+g(x,u,Du,D2u),in[0,∞)×Ω,u(0,x)=φ,ut(0,x)=ψ,inΩ,u(t,x)=0,on⁡[0,∞)×∂Ω,
where
(1.2)ut=∂u∂t,utt=∂2u∂t2,Δ=∑i=1n∂2∂xi2,D=(∂∂x1,…,∂∂xn),D2=∂2∂x1α1⋯∂xnαn,α1+⋯+αn=2,x=(x1,…,xn),k>0.
Equations of type (1.1) are a class essential nonlinear wave equations describing the speed of strain waves in a viscoelastic configuration (e.g., a bar if the space dimension N=1 and a plate if N=2) made up of the material of the rate type [1, 2]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model [3]. Concerning damped cases, there is much to the global existence of solutions for the problem:
(1.3)utt+ut-uxx=f(u),in[0,∞)×Ω,u(0,x)=u0(x),ut(0,x)=u1(x),inΩ;
they discussed the global existence of weak solutions and regularity in R1 and Rn [4–8]. On the other hand, Ikehata and Inoue [9] considered the global existence of weak solutions for two-dimensional problem in an exterior domain Ω⊂R2 with a compact smooth boundary ∂Ω for a semilinear strongly damped wave equation with a power-type nonlinearity |u|q and q>6:
(1.4)utt(t,x)-Δu(t,x)-Δut(t,x)=|u(t,x)|qin[0,∞)×Ω,u(0,x)=u0(x),ut(0,x)=u1(x)inΩ,u(t,x)=0,on[0,∞)×∂Ω.
Cholewa and Dlotko [10] discussed the global solvability and asymptotic behavior of solutions to semilinear Cauchy problem for strongly damped wave equation in the whole of Rn. They assume the nonlinear term f grows like |u|q and q<(n+2)/(n-2) if n≥3. Similar problems attracted attention of the researchers for many years [11–13]. Especially, Yang [14] studied the global existence of weak solutions to the more general equation including (1.4), but he did not discuss the regularity of weak solution for the quasilinear wave equation. We are interested in discussing the global existence and regularity of weak solutions for strongly damped wave equation with the dissipative terms g containing Du and the nonlinear terms f containing Δu. Here f(x,Δu) is a given monotone in Δu nonlinearity satisfying some dissipativity and growth restrictions and g(x,u,Du,D2u) is in a sense subordinated to f(x,Δu).In [15], we have investigated the existence of global solutions to a class of nonlinear damped wave operator equations. In this paper, our first aim is to study the global existence of strong solutions to the more general equation including (1.4), which is the motivation that we establish our abstract strongly damped wave equation model with. The second aim is to deal with the global existence of strong solutions to a class of fully nonlinear wave equations with strongly damped terms under some weakly growing conditions.This paper is organized as follows:(i)
in Section2, we recall some preliminary tools and definitions;(ii)
in Section3, we put forward our abstract strongly damped wave equation model and proof the global existence of strong solution of it;(iii)
in Section4, we provide the proof of the main results about the mixed problem (1.1).


## 2. Preliminaries

We introduce two spatial sequences:(2.1)X⊂H3⊂X2⊂X1⊂H,X2⊂H2⊂H1⊂H,
where H, H1, H2, and H3 are Hilbert spaces, X is a linear space, and X1,X2 are Banach spaces.All embeddings of (2.1) are dense. Let
(2.2)L:X⟶X1beone-for-onedenselinearoperator,〈Lu,v〉H=〈u,v〉H1,∀u,v∈X.Furthermore,L has eigenvectors {ek} satisfying
(2.3)Lek=λkek,(k=1,2,…),
and {ek} constitutes common orthogonal basis of H and H3.We consider the following abstract wave equation model:(2.4)d2udt2+kddtLu=G(u),k>0,u(0)=φ,ut(0)=ψ,
where G:X2×R+→X1* is a map, R+=[0,∞), and ℒ:X2→X1 is a bounded linear operator, satisfying
(2.5)〈Lu,Lv〉H=〈u,v〉H2,∀u,v∈X2.Definition 2.1.
We say thatu∈W1,∞((0,T),H1)∩L∞((0,T),X2) is a global weak solution of the (2.4) provided for (φ,ψ)∈X2×H1(2.6)〈ut,v〉H+k〈Lu,v〉H=∫0t‍〈G(u),v〉dτ+〈ψ,v〉H+k〈Lφ,v〉H,
for each v∈X1 and 0≤t≤T<∞.Definition 2.2.
Letun, u0∈Lp((0,T),X2). We say that un⇀u0 in Lp((0,T),X2) is uniformly weakly convergent if {un}⊂L∞((0,T),H) is bounded, and
(2.7)un⇀u0,inLp((0,T),X2),lim⁡n→∞∫0T‍|〈un-u0,v〉H|2dt=0,∀v∈H.Lemma 2.3 (see [16]).
Let{un}∈Lp((0,T),Wm,p(Ω))(m≥1) be bounded sequences and {un} uniformly weakly convergent to u0∈Lp((0,T),Wm,p(Ω)). Then, for each |α|≤m-1, it follows that
(2.8)Dαun⟶Dαu0,inL2((0,T)×Ω).Lemma 2.4 (see [17]).
LetΩ⊂Rn be an open set and f:Ω×RN→R1 satisfy Caratheodory condition and
(2.9)|f(x,ξ)|≤C∑i=1N‍|ξi|pi/p+b(x).
If {uik}⊂Lpi(Ω)(1≤i≤N) is bounded and uik convergent to ui in Ω0 for all bounded Ω0⊂Ω, then for each v∈Lp′(Ω), the following equality holds
(2.10)lim⁡k→∞∫Ω‍f(x,u1k,…,uNk)vdx=∫Ω‍f(x,u1,…,uN)vdx.


## 3. Model Results

LetG=A+B:X2×R+→X1*. Assume(A1)
there is aC1 functional F:X2→R1 such that
(3.1)〈Au,Lv〉=〈-DF(u),v〉,∀u,v∈X;(A2)
functionalF:X2→R1 is coercive, that is,
(3.2)F(u)⟶∞,⟺‖u‖X2⟶∞;(A3)
B  satisfies
(3.3)|〈Bu,Lv〉|≤C1F(u)+k2‖v‖H12+C2,∀u,v∈X,forg∈Lloc1(0,∞).Theorem 3.1.
SetG:X2×R+→X1*, for each (φ,ψ)∈X2×H1, then the following assertions hold.(1)
IfG=A satisfies (A1) and (A2), then (2.4) has a globally weak solution
(3.4)u∈W1,∞((0,∞),H1)∩W1,2((0,∞),H2)∩L∞((0,∞),X2).(2)
IfG=A+B satisfies (A1)–(A3), then (2.4) has a global weak solution
(3.5)u∈Wloc⁡1,∞((0,∞),H1)∩Wloc⁡1,2((0,∞),H2)∩Lloc⁡∞((0,∞),X2).(3)
Furthermore, ifℒ:X2→X1 is symmetric sectorial operator, that is, 〈ℒu,v〉=〈u,ℒv〉, and G=A+B satisfies
(3.6)|〈Gu,v〉|≤C1F(u)+12‖v‖H2+C2,then u∈Wloc2,2((0,∞),H).Proof.
Let{ek}⊂X be a common orthogonal basis of H and H3, satisfying (2.3). Set
(3.7)Xn={∑i=1n‍αiei∣αi∈R1},X~n={∑j=1n‍βj(t)ej∣βj∈C2[0,∞)}.
Clearly, LXn=Xn, LX~n=X~n.
By using Galerkin method, there exitsun∈C2([0,∞),Xn) satisfying
(3.8)〈dundt,v〉H+k〈Lun,v〉H=∫0t‍〈G(un),v〉dτ+〈ψn,v〉H+k〈Lφn,v〉H,un(0)=φn,un′(0)=ψn,
for  ∀v∈Xn, and
(3.9)∫0t‍[〈d2undt2,v〉H+k〈Ldundt,v〉H]dτ=∫0t‍〈Gun,v〉dτ
for ∀v∈X~n.
Firstly, we considerG=A. Let v=(d/dt)Lun in (3.9). Taking into account (2.2) and (3.1), it follows that
(3.10)0=∫0t‍[〈d2undt2,ddtLun〉]+k〈Ldundt,ddtLun〉Hdτ-∫0t‍〈Aun,ddtLun〉dτ,0=∫0t‍[12ddt〈dundt,dundt〉H1+k〈dundt,dundt〉H2+〈DF(un),dundt〉]dτ=12‖dundt‖H12-12‖ψn‖H12+k∫0t‍‖dundt‖H22dτ+F(un)-F(φn).
We get(3.11)12‖dundt‖H12+k∫0t‍‖dundt‖H22dτ+F(un)=F(φn)+12‖ψn‖H12.
Letφ∈H3. From (2.1) and (2.2), it is known that {en} are orthogonal basis of H1. We find that φn→φ in H3, and ψn→ψ in H1. Since H3⊂X2 is imbedding, it follows that
(3.12)φn⟶φ,inX2,ψn⟶ψ,inH1.
From (3.2), (3.11), and (3.12), we obtain that,
(3.13){un}⊂Wloc⁡1,∞((0,∞),H1)∩Wloc⁡1,2((0,∞),H2)∩Lloc⁡∞((0,∞),X2)isbounded.
Let(3.14)un*⇀u0,inWloc⁡1,∞((0,∞),H1)∩Lloc⁡∞((0,∞),X2),un⇀u0,inWloc⁡1,2((0,∞),H2),
which implies that un→u0 in Wloc1,2((0,∞),H) is uniformly weakly convergent from that H2⊂H is compact imbedding.
If we have the following equality:(3.15)lim⁡n→∞[-∫0t|〈Gun-Gu0,Lun-Lu0〉|dτ+k2‖un-u0‖H22]=0,
then u0 is a weak solution of (2.4) in view of (3.8), (3.14).
We will show (3.15) as follows. It follows that from (2.5),
(3.16)∫0t‍〈ddtLun-ddtLu0,Lun-Lu0〉Hdτ=12∫0t‍ddt〈un-u0,un-u0〉H2dτ=12‖un(t)-u0(t)‖H22-12‖φn-φ‖H22.
Taking into account (2.2), (2.5), and (3.9), we get that
(3.17)-∫0t‍〈Gun-Gu0,Lun-Lu0〉dτ+k2‖un-u0‖H22=∫0t‍[〈Gu0-Gun,Lun-Lu0〉+k〈ddtLun-ddtLu0,Lun-Lu0〉H]dτ+k2‖φn-φ‖H22=∫0t‍[〈Gu0,Lun-Lu0〉+〈Gun,Lu0〉-〈Gun,Lun〉-k〈dundt,u0〉H2-k〈du0dt,un-u0〉H2+k〈ddtLun,Lun〉H]dτ+k2‖φn-φ‖H22=∫0t‍[〈Gu0,Lun-Lu0〉+〈Gun,Lu0〉-k〈dundt,u0〉H2-k〈du0dt,un-u0〉H2-〈d2undt2+kddtLun,Lun〉H+k〈ddtLun,Lun〉H]dτ+k2‖φn-φ‖H22=∫0t‍[〈dundt,dundt〉H1〈Gu0,Lun-Lu0〉+〈Gun,Lu0〉-k〈dundt,u0〉H2-k〈ddtu0,un-u0〉H2+〈dundt,dundt〉H1]dτ-〈dundt,un〉H1+〈ψn,φn〉H1+k2‖φn-φ‖H22.
From (2.1) and (3.14), we have
(3.18)lim⁡n→∞‖φn-φ‖H2=0,lim⁡n→∞∫0t‍〈Gu0,Lun-Lu0〉dτ=0,lim⁡n→∞∫0t‍〈ddtu0,un-u0〉H2dτ=0.
Then, we get(3.19)lim⁡n→∞-∫0t‍〈Gun-Gu0,Lun-Lu0〉dτ+k2lim⁡n→∞‖un-u0‖H22=lim⁡n→∞∫0t‍[〈Gun,Lu0〉-k〈dundt,u0〉H2+‖dundt‖H12]dτ-lim⁡n→∞〈dundt,un〉H1+〈ψ,φ〉H1.
In view of (3.9), (3.14), we obtain for all v∈∪n=1∞X~n(3.20)lim⁡n→∞∫0t‍〈Gun,Lv〉dτ=∫0t‍[k〈du0dt,v〉H2-〈du0dt,dvdt〉H1]dτ+〈du0dt,v〉H1-〈ψ,v(0)〉H1.
Since∪n=1∞X~n is dense in W1,2((0,T),H2)∩Lp((0,T),X2), for all p<∞, (3.20) holds for all v∈W1,2((0,T),H2)∩Lp((0,T),X2). Thus, we have
(3.21)lim⁡n→∞∫0t‍〈Gun,Lu0〉dτ=∫0t‍[k〈du0dt,u0〉H2-‖du0dt‖H12]dτ+〈du0dt,u0〉H1-〈ψ,φ〉H1.
From (3.14) and H2⊂H1 being compact imbedding, it follows that
(3.22)lim⁡n→∞∫0t‍‖dundt‖H12dτ=∫0t‍‖du0dt‖H12dτ,lim⁡n→∞〈dundt,un〉H1=〈du0dt,u0〉H1,a.e.t≥0.
Clearly,(3.23)lim⁡n→∞∫0t〈dundt,un〉H1dτ=∫0t〈du0dt,u0〉H1dτ.
Then, (3.14) follows from (3.19)–(3.21), which implies assertion (1).
Secondly, we considerG=A+B. Let v=(d/dt)Lun in (3.9). In view of (2.2) and (2.8), it follows that
(3.24)∫0t‍[〈d2undt2,ddtLun〉H+k〈Ldundt,ddtLun〉H]dτ=∫0t‍〈(A+B)un,ddtLun〉Hdτ∫0t‍[12ddt〈dundt,dundt〉H1+k〈dundt,dundt〉H2]dτ=∫0t‍[〈-DF(un),dundt〉+〈B(un),ddtLun〉H]dτ,12‖dundt‖H12-12‖ψn‖H12+k∫0t‍‖dundt‖H22dτ+F(un)-F(φn)=∫0t‍〈B(un),ddtLun〉Hdτ,12‖dundt‖H12+k∫0t‍‖dundt‖H22dt+F(un)=∫0t‍〈B(un),ddtLun〉dτ+F(φn)+12‖ψn‖H12.
From (3.3), we have
(3.25)12‖dundt‖H12+F(un)+k∫0t‍‖dundt‖H22dτ≤C∫0t‍[C1F(un)+k2‖dundt‖H12+C2]dτ+F(φn)+12‖ψn‖H12≤C∫0t‍[F(un)+12‖dundt‖H12]dτ+f(t),
where f(t)=(1/2)∥ψ∥H12+supnF(φn).
By using Gronwall inequality, it follows that(3.26)12‖dundt‖H12+F(un)≤f(0)eCt+∫0t‍f(τ)eC(t-τ)dτ,
which implies that for all 0<T<∞,
(3.27){un}⊂W1,∞((0,T),H1)∩L∞((0,T),X2)isbounded.
From (3.25) and (3.21), it follows that
(3.28){un}⊂W1,2((0,T),H2)isbounded.
Let(3.29)un*⇀u0,inW1,∞((0,T),H1)∩L∞((0,T),X2),un⇀u0,inW1,2((0,T),H2),
which implies that un→u0 in W1,2((0,T),H) is uniformly weakly convergent from that H2⊂H is compact imbedding.
The left proof is same as assertion (1).
Lastly, assume (3.6) hold. Let v=d2un/dt2 in (3.9). It follows that
(3.30)∫0t[〈d2undt2,d2undt2〉H+k〈Ldundt,d2undt2〉H]dτ=∫0t〈(A+B)(un),d2undt2〉dτ≤∫0t[CF(un)+12‖d2undt2‖H2+g(t)]dτ,∫0t[〈d2undt2,d2undt2〉H+k2∫0t‍∫Ω‍ddt(un′(t))2]dxdτ≤∫0t[CF(un)+12‖d2undt2‖H2+g(t)]dτ,∫0t〈d2undt2,d2undt2〉Hdt+k2‖dundt‖H12≤k2‖ψn‖H2+∫0t‍[12‖d2undt2‖H2+CF(un)+g(τ)]dτ.
From (3.26), the above inequality implies
(3.31)∫0t‍‖d2undt2‖H2dτ≤C,(C>0isconstant).
We see that for all 0<T<∞, {un}⊂W2,2((0,T),H) is bounded. Thus u∈W2,2((0,T),H).


## 4. Main Result

Now, we begin to consider the mixed problem (1.1). Set
(4.1)F(x,y)=∫0y‍f(x,z)dz.
We assume
(4.2)F(x,y)≥C1|y|p-C2,p≥2,|f(x,y)|≤C(|y|p-1+1),(4.3)(f(x,y1)-f(x,y2))(y1-y2)≥λ|y1-y2|2,λ>0,(4.4)|g(x,z,ξ,η)|≤C(|z|p/2+|ξ|p/2+|η|p/2+1),(4.5)|g(x,z,ξ,η1)-g(x,z,ξ,η2)|≤K1|η1-η2|,
where C,C1,C2 are constant and K1<λK, K is the best constant satisfying
(4.6)K2‖u‖H22≤∫Ω‍|Δu|2dx.Theorem 4.1.
If the assumptions of (4.1)–(4.5) hold, for (φ,ψ)∈W2,p(Ω)∩W01,p(Ω)×H01(Ω), then (1.1) is a strong solution
(4.7)u∈Lloc⁡∞((0,∞),W2,p(Ω)∩W01,p(Ω)),ut∈Lloc⁡∞((0,∞),H01(Ω))∩Lloc⁡2((0,∞),H2(Ω)),utt∈Lp′((0,T)×Ω),p′=pp-1,∀0<T<∞.Proof.
We introduce spatial sequences(4.8)X={u∈C∞(Ω)|Δku|∂Ω=0,k=0,1,2,…},X1=Lp(Ω),X2=W2,p(Ω)∩W01,p(Ω),H=L2(Ω),H1=H01(Ω),H2=H2(Ω)∩H01(Ω),H3={u∈H2m(Ω):u|∂Ω=⋯=Δm-1u|∂Ω=0},
where the inner products of H2 and H3 are defined by
(4.9)〈u,v〉H2=∫Ω‍ΔuΔvdx,〈u,v〉H2=∫Ω‍ΔmuΔmvdx,
where m≥1 such that H3⊂X2 is an embedding.
Linear operatorsℒ:X→X1 and L:X→X1 are defined by
(4.10)Lu=Lu=-Δu.
It is known that ℒ and L satisfy (2.2), (2.3), and (2.5). Define G=A+B:X2→X1* by
(4.11)〈Au,v〉=∫Ω‍f(x,Δu)vdx,〈Bu,v〉=∫Ω‍g(x,u,Du,D2u)vdx,forv∈X1.
We show thatG=A+B:X2→X1* is T-coercively weakly continuous. Let {un}⊂L∞((0,T),W2,p(Ω)∩W01,p(Ω)) satisfying (2.7) and
(4.12)lim⁡n→∞∫0T‍|〈Gun-Gu0,Lun-Lu0〉|dt=lim⁡n→∞∫0T‍∫Ω‍[(f(x,Δun)-f(x,Δu0))(un-u0)+(g(x,un,Dun,D2un)-g(x,u0,Du0,D2u0))(un-u0)]dxdt=0.
We need to prove that(4.13)lim⁡n→∞∫0T‍∫Ω‍[f(x,Δun)+g(x,un,Dun,D2un)]vdxdt=∫0T‍∫Ω‍[f(x,Δu0)+g(x,u0,Du0,D2u0)]vdxdt.
From (2.7) and Lemma 2.3, we obtain
(4.14)un⟶u0,Dun⟶Du0inL2((0,T)×Ω).
From (4.3), we get
(4.15)∫0T‍∫Ω‍[f(x,Δun)-f(x,Δu0)](Δun-Δu0)dxdt≥λ∫0T‍∫Ω‍|Δun-Δu0|2dxdt.
We have the deformation(4.16)∫0T‍∫Ω‍[g(x,un,Dun,D2un)-g(x,u0,Du0,D2u0)](Δun-Δu0)dxdt=∫0T‍∫Ω‍[g(x,un,Dun,D2u0)-g(x,u0,Du0,D2u0)](Δun-Δu0)dxdt+∫0T‍∫Ω‍[g(x,un,Dun,D2un)-g(x,un,Dun,D2u0)](Δun-Δu0)dxdt.
From (4.14) and Lemma 2.4, we have
(4.17)lim⁡n→∞∫0T‍∫Ω‍[g(x,un,Dun,D2u0)-g(x,u0,Du0,D2u0)](Δun-Δu0)dxdt=0.
From (4.12), (4.15)–(4.17), it follows that
(4.18)0≥λ∫0T‍∫Ω‍|Δun-Δu0|2dxdt+∫0T‍∫Ω‍[g(x,un,Dun,D2un)-g(x,un,Dun,D2u0)]×(Δun-Δu0)dxdt≥λ∫0T‍∫Ω‍|Δun-Δu0|2dxdt-K1∫0T‍∫Ω‍|D2un-D2u0||Δun-Δu0|dxdt≥λ2∫0T‍∫Ω‍|Δun-Δu0|2dxdt-K122λ∫0T‍∫Ω‍|D2un-D2u0|2dxdt≥λ2K2-K122λ∫0T‍∫Ω‍|D2un-D2u0|2dxdt.
SinceλK>K1, we have
(4.19)lim⁡n→∞∫0T‍∫Ω‍|D2un-D2u0|2dxdt=0.
From (4.14), (4.19), (4.1), (4.4), and Lemma 2.3, we get (4.13).
LetF1(u)=∫Ω‍F(x,Δu)dx, where F is same as (4.1). We get
(4.20)〈Au,Lu〉=-〈DF1(u),v〉,F(u)⟶∞⟺‖u‖X2⟶∞,
which implies Conditions (A1), (A2) of model results in Theorem 3.1.
We will show (3.3) as follows. It follows that
(4.21)|〈Bu,Lv〉|=∫Ω‍|g(x,u,Du,D2u)||Δv|dx≤k2∫Ω‍|Δv|2dx+2k∫Ω‍|g(x,u,Du,D2u)|2dx≤k2‖v‖H22+C∫Ω‍[|D2u|p+|∇u|p+|u|p+1]dx≤k2‖v‖H22+CF1(u)+C,
which imply Conditions (A3) of Theorem 3.1. From Theorem 3.1, (1.1) has a solution
(4.22)u∈Lloc⁡∞((0,∞),W2,p(Ω)∩W01,p(Ω)),ut∈Lloc⁡∞((0,∞),H01(Ω))∩Lloc⁡2((0,∞),H2(Ω)),utt∈Lp′((0,T)×Ω),p′=pp-1,∀0<T<∞,
satisfying
(4.23)∫Ω‍∂u∂tvdx-k∫Ω‍Δuvdx=∫0t‍∫Ω‍f(x,Δu)vdxdτ+∫0t‍∫Ω‍g(x,u,Du,D2u)vdxdτ+∫Ω‍ψvdx-k∫Ω‍Δφvdx.



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*Source: 805158-2012-07-12.xml*