# The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

Authors: Meng Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev spaceC([0,T);Hs(R))∩C1([0,T);Hs-1(R)) provided that the initial value lies in Hs(R) with s>3/2. Meanwhile, we prove the existence of global weak solutions in L∞([0,∞);L2(R)) for the equation.


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## Body

## 1. Introduction

Coclite and Karlsen [1] investigated the well posedness in classes of discontinuous functions for the generalized Degasperis-Procesi equation:
(1)ut-utxx+4h′(u)ux=h′′′(u)ux3+3h′′(u)uxuxx+h′(u)uxxx,
which is subject to the condition
(2)|h′(u)|≤c|u|,|h(u)|≤c|u|2,
or
(3)|h′(u)|≤c,|h(u)|≤c|u|,
where c is a positive constant. The existence and L1 stability of entropy weak solutions belonging to the class L1(R)∩BV(R) are established for (1) in paper [1].In this work, we study the following model:(4)ut-utxx+mh′(u)ux=h′′′(u)ux3+3h′′(u)uxuxx+h′(u)uxxx,
where m is a positive constant and h(u)∈C3. If m=4 and h(u)=u2/2, (4) reduces to the classical Degasperis-Procesi model (see [2–13]). Here, we notice that assumptions (2) and (3) do not include the case h(u)=u3. In this paper, we will study the case h(u)=u3, and m is an arbitrary positive constant.In fact, the Cauchy problem of (4) in the case h(u)=u3 is equivalent to the following system:
(5)ut-utxx+3mu2ux=6ux3+18uuxuxx+3u2uxxx,u(0,x)=u0(x).
Using the operator (1-∂x2)-1 to multiply the first equation of the problem (5), we obtain
(6)ut+3u2ux+(m-1)(1-∂x2)-1∂x(u3)=0,u(0,x)=u0(x).It is shown in this work that there exists a unique local strong solution in the Sobolev spaceC([0,T);Hs(R))∩C1([0,T);Hs-1(R)) by assuming that the initial value u0(x) belongs to Hs(R) with s>3/2. In addition, we prove the existence of global weak solutions in L∞([0,∞);L2(R)) for the system (6).This paper is organized as follows. Section2 investigates the existence and uniqueness of local strong solutions. The result about global weak solution is given in Section 3.


## 2. Local Existence

In this section, we will use the Kato theorem in [14] for abstract differential equation to establish the existence of local strong solution for the problem (6). Let us consider the following problem:
(7)dvdt+H(v)v=g(v),t≥0,v(0)=v0.
Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let Q:Y→X be a topological isomorphism. Let L(Y,X) be the space of all bounded linear operators from Y to X. In the case of X=Y, we denote this space by L(X). We illustrate the following conditions in which σ1, σ2, σ3, and σ4 are constants depending only on max{∥y∥Y,∥z∥Y}.(i)
H(y)∈L(Y,X) for y∈X with
(8)∥(H(y)-H(z))w∥X≤σ1∥y-z∥X∥w∥Y,y,z,w∈Y,
and H(y)∈G(X,1,β) (i.e., H(y) is quasi-m-accretive), uniformly on bounded sets in Y.(ii)
QH(y)Q-1=H(y)+A(y), where A(y)∈L(X) is bounded, uniformly on bounded sets in Y. Moreover,
(9)∥(A(y)-A(z))w∥X≤σ2∥y-z∥Y∥w∥X,y,z∈Y,w∈X.(iii)
g:Y→Y extends to a map from X into X, is bounded on bounded sets in Y, and satisfies
(10)∥g(y)-g(z)∥Y≤σ3∥y-z∥Y,y,z∈Y,(11)∥g(y)-g(z)∥X≤σ4∥y-z∥X,y,z∈X.Kato Theorem (see [14]).
Assume that conditions (i), (ii), and (iii) hold. Ifv0∈Y, there is a maximal T>0 depending only on ∥v0∥Y and a unique solution v to the problem (7) such that
(12)v=v(·,v0)∈C([0,T);Y)∩C1([0,T);X).
Moreover, the map v0→v(·,v0) is a continuous map from Y to the following space:
(13)C([0,T);Y)∩C1([0,T);X).In order to apply the Kato theorem to establish the local well posedness for the problem (6), we let H(u)=3u2∂x, Y=Hs(R), X=Hs-1(R), Λ=(1-∂x2)1/2, g(u)=(m-1)Λ-2∂x(u3), and Q=Λs. We know that Q is an isomorphism of Hs onto Hs-1. Now, we cite the following Lemmas.Lemma 1.
The operatorA(u)=u2∂x with u∈Hs(R), s>3/2 belongs to G(Hs-1(R),1,β).Lemma 2.
Assume thatH(u)=3u2∂x with u∈Hs(R) and s>3/2. Then, H(u)∈L(Hs(R),Hs-1(R)) for all u∈Hs(R). Moreover,
(14)∥(H(u)-H(z))w∥Hs-1≤σ1∥u-z∥Hs-1∥w∥Hs,u,z,w∈Hs(R).Lemma 3.
Fors>3/2, u,z∈Hs(R) and w∈Hs-1, it holds that A(u)=[Λs,3u2∂x]Λ-s∈L(Hs-1) for u∈Hs and
(15)∥(A(u)-A(z))w∥Hs-1≤σ2∥u-z∥Hs∥w∥Hs-1.The above three Lemmas can be found in Ni and Zhou [15].Lemma 4.
Letu,z∈Hs with s>3/2 and g(u)=(m-1)Λ-2∂x(u3). Then, g is bounded on bounded sets in Hs and satisfies
(16)∥g(u)-g(z)∥Hs≤σ3∥u-z∥Hs,∥g(u)-g(z)∥Hs-1≤σ4∥u-z∥Hs-1.Proof.
Fors0>1/2, we know that ∥uv∥Hs0(R)≤c∥u∥Hs0(R)∥v∥Hs0(R). Consequently, we have
(17)∥g(u)-g(z)∥Hs≤c∥u3-z3∥Hs-1≤c∥u-z∥Hs-1(∥u∥Hs-12+∥v∥Hs-12)≤σ3∥u-z∥Hs,∥g(u)-g(z)∥Hs-1≤c∥u3-z3∥Hs-2≤c∥u3-z3∥Hs-1≤c∥u-z∥Hs-1(∥u∥Hs-12+∥v∥Hs-12)≤σ4∥u-z∥Hs-1.Using the Kato Theorem, Lemmas1–4, we immediately obtain the local well-posedness theorem.Theorem 5.
Assume thatu0∈Hs(R) with s>3/2. Then, there exists a T>0 such that the system (5) or the problem (6) has a unique solution u(t,x) satisfying
(18)u(t,x)∈C([0,T);Hs(R))∩C1([0,T);Hs-1(R)).


## 3. Weak Solutions

In this section, our aim is to establish the existence of global weak solutions for the system (6). Firstly, we prove that the solution of the problem (5) is bounded in the space L2(R) and L∞(R).Lemma 6.
The solution of the problem (5) with m>0 satisfies
(19)∫R‍K1Kdx=∫R‍1+ξ2m+ξ2|u^(ξ)|2dξ=∫R‍1+ξ2m+ξ2|u^0(ξ)|2dξ,
where K1=u-∂xx2u, and K=(m-∂xx2)-1u. Moreover, there exist two constants c1>0 and c2>0 depending only on m such that
(20)c1∥u0∥L2(R)≤c1∥u∥L2(R)≤c2∥u0∥L2(R).Proof.
SettingK1=u-∂xx2u and K=(m-∂xx2)-1u and using the first equation of the problem (5), we obtain u=my-yxx and
(21)ddt∫R‍K1Kdx=∫R‍∂K1∂tKdx+∫R‍K1∂K∂tdx=2∫R‍∂K1∂tKdx=2∫R‍[-3mu2ux+6ux3+18uxuxx+3u2uxxx]Kdx=2∫R‍[-m∂x(u3)+(u3)xxx]Kdx=∫R‍(mu3)Kx-u3Kxxxdx=∫R‍[mu3]Kx-u3(mKx-ux)dx=∫R‍u3uxdx,=0.
Using the Parseval identity and (21), we obtain (19) and (20).From Theorem5, we know that for any u0∈Hs(R) with s>3/2, there exists a maximal T=T(u0)>0 and a unique strong solution u to the problem (6) such that
(22)u∈C([0,T);Hs(R))∩C1([0,T);Hs-1(R)).
Firstly, we study the following differential equation:
(23)pt=3u2(t,p),t∈[0,T),p(0,x)=x.Lemma 7.
Letu0∈Hs, s>3, and let T>0 be the maximal existence time of the solution to the problem (6). Then, the problem (23) has a unique solution p∈C1([0,T)×R,R). Moreover, the map p(t,·) is an increasing diffeomorphism of R with px(t,x)>0 for (t,x)∈[0,T)×R.Proof.
Using Theorem5, we obtain u∈C1([0,T);Hs-1(R)) and Hs-1∈C1(R). Therefore, we know that functions u(t,x) and ux(t,x) are bounded, Lipschitz in space, and C1 in time. Using the existence and uniqueness theorem for ordinary differential equations derives that the problem (23) has a unique solution p∈C1([0,T)×R,R).
Differentiating (23) with respect to x gives rise to the following:
(24)ddtpx=6uux(t,p)px,t∈[0,T),px(0,x)=1,
from which we obtain
(25)px(t,x)=exp(∫0t‍6uux(τ,p(τ,x))dτ).
For every T′<T, using the Sobolev imbedding theorem yields that
(26)sup(τ,x)∈[0,T′)×R|ux(τ,x)|<∞.
It is inferred that there exists a constantK0>0 such that px(t,x)≥e-K0t for (t,x)∈[0,T)×R. It completes the proof.Lemma 8.
Assume thatu0∈Hs(R),s>3/2. Let T be the maximal existence time of the solution u to the problem (6). Then, it has
(27)∥u(t,x)∥L∞≤∥u0∥L∞ect∀t∈[0,T],
where c>0 is a constant independent of t.Proof.
Letξ(x)=(1/2)e-|x|, we have (1-∂x2)-1g=ξ⋆f for all g∈L2(R) and u=ξ⋆K1(t,x). Using a simple density argument presented in [7], it suffices to consider s=3 to prove this lemma. Let T be the maximal existence time of the solution u to the problem (6) with the initial value u0∈H3(R) such that u∈C([0,T),H3(R))∩C1([0,T),H2(R)). From (6), we have
(28)ut+3u2ux=-(m-1)ξ⋆(3u2ux).
Since
(29)-ξ⋆(3u2ux)=-12∫-∞∞‍e-|x-η|3u2uηdη=-32∫-∞x‍e-x+ηu2uηdη-32∫x+∞‍ex-ηu2uηdη=12∫∞x‍e-|x-η|u3dη-12∫x∞‍e-|x-η|u3dη,du(t,p(t,x))dt=ut(t,p(t,x))+ux(t,p(t,x))dp(t,x)dt=(ut+3u2ux)(t,p(t,x)),
from (29), we have
(30)du(t,p(t,x))dt=m-12∫-∞p(t,x)‍e-|p(t,x)-η|u3dη-m-12∫p(t,x)∞‍e-|p(t,x)-η|u3dη.
Using Lemma 6 and (30) derives that
(31)|du(t,p(t,x))dt|≤|m-1|2∫-∞∞‍e-|p(t,x)-η|u2dη≤|m-1|2∫-∞∞‍u3dη≤|m-1|2∥u∥L22∥u∥L∞≤c∥u0∥L2(R)∥u∥L∞≤c∥u∥L∞,
where c is a positive constant independent of t. Using (31) results in the following:
(32)-c∫0t∥u∥L∞(R)‍dt+u0≤u(t,p(t,x))≤c∫0t‍∥u∥L∞(R)dt+u0.
Therefore,
(33)|u(t,p(t,x))|≤∥u(t,p(t,x))∥L∞≤∥u0∥L∞+c∫0t‍∥u∥L∞(R)dt.
Using the Sobolev embedding theorem to ensure the uniform boundedness of ux(s,η) for (s,η)∈[0,t]×R with t∈[0,T′), from Lemma 7, for every t∈[0,T′), we get a constant C(t) such that
(34)e-C(t)≤px(t,x)≤eC(t),x∈R.
We deduce from (34) that the function p(t,·) is strictly increasing on R with limx→±∞p(t,x)=±∞ as long as t∈[0,T′). It follows from (33) that
(35)∥u(t,x)∥L∞=∥u(t,p(t,x))∥L∞≤∥u0∥L∞+c∫0t‍∥u∥L∞(R)dt.
Using the Gronwall inequality and (35) derives that (27) holds.For a real numbers with s>0, suppose that the function u0(x) is in Hs(R), and let uε0 be the convolution uε0=ϕε⋆u0 of the function ϕε(x)=ε-1/4ϕ(ε-1/4x) and u0 such that the Fourier transform ϕ^ of ϕ satisfies ϕ^∈C0∞, ϕ(ξ)^≥0, and ϕ(ξ)^=1 for any ξ∈(-1,1). Then, we have uε0(x)∈C∞. It follows from Theorem 5 that for each ε satisfying 0<ε<1/2, the Cauchy problem,
(36)ut-utxx+3mu2ux=6ux3+18uuxuxx+3u2uxxx,u(0,x)=uε0(x),
has a unique solution uε(t,x)∈C∞([0,T);H∞). Using Lemmas 6 and 8, for every t∈[0,T), we obtain
(37)∥uε(t,x)∥L2(R)≤c∥uε(0,x)∥L2(R)≤c∥u0∥L2(R),∥uε(t,x)∥L∞≤∥uε(0,x)∥L∞ect≤c∥u0∥L∞ect.
Sending t→T, we know that inequalities (37) are still valid. This means that for t∈[0,∞), (37) hold.Now, we state the concepts of weak solutions.Definition 9 (weak solution).
We call a functionu:R+×R→R a weak solution of the Cauchy problem (5) provided that(i)
u∈L∞(R+;L2(R));(ii)
ut-utxx+3mu2ux=6ux3+18uuxuxx+3u2uxxx in D′([0,∞)×R), that is, for all φ∈C0∞([0,∞)×R) there holds the following identity:
(38)∫R+‍∫R‍(u(φt-φtxx)+mu3φx-u3φxxx)dxdt+∫R‍u0(x)φ(0,x)dx=0.Theorem 10.
Letu0(x)∈L2(R). Then, there exists a weak solution u(t,x)∈L∞([0,∞);L2(R)) to the problem (5).Proof.
Consider the problem (36). For an arbitrary T>0, choosing a subsequence εn→0, from (37), we know that uεn is bounded in L∞ and ∥uεn∥L2(R) is uniformly bounded in L2(R). Therefore, we obtain that uεn3 is bounded in L2(R). Therefore, there exist subsequences {uεn} and {uεn3}, still denoted by {uεn} and {uεn3}, are weakly convergent to v in L2(R). Noticing (38) completes the proof.



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*Source: 619068-2013-05-20.xml*