Li WEI(魏利)Rui CHEN(陳蕊)

School of Mathematics and Statistics,Hebei University of Economics and Business, Shijiazhuang 050061,China

Ravi P.AGARWAL

Department of Mathematics,Texas A&M University-Kingsville,Kingsville,TX 78363,USA; Department of Mathematics,Faculty of Science,King Abdulaziz University, 21589 Jeddah,Saudi Arabia

Patricia YJ WONG

School of Electrical and Electronic Engineering,Nanyang Technological University, 50 Nanyang Avenue,Singapore 639798,Singapore

EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES?

Li WEI(魏利)Rui CHEN(陳蕊)

School of Mathematics and Statistics,Hebei University of Economics and Business, Shijiazhuang 050061,China

E-mail:diandianba@yahoo.com;stchenri@heuet.edu.cn

Ravi P.AGARWAL

Department of Mathematics,Texas A&M University-Kingsville,Kingsville,TX 78363,USA; Department of Mathematics,Faculty of Science,King Abdulaziz University, 21589 Jeddah,Saudi Arabia

E-mail:Ravi.Agarwal@tamuk.edu

Patricia YJ WONG

School of Electrical and Electronic Engineering,Nanyang Technological University, 50 Nanyang Avenue,Singapore 639798,Singapore

E-mail:ejywong@ntu.edu.sg

One parabolic p-Laplacian-like diferential equation with mixed boundaries is studied in this paper,where the itemin the corresponding studies is replaced bywhich makes it more general.The sufcient condition of the existence and uniqueness of non-trivial solution in L2(0,T;L2(?))is presented by employing the techniques of splitting the boundary problems into operator equation.Compared to the corresponding work,the restrictions imposed on the equation are weaken and the proof technique is simplifed.It can be regarded as the extension and complement of the previous work.

maximal monotone operator;Caratheodory’conditions;subdiferential; p-Laplacian-like equation;nontrivial solution

2010 MR Subject Classifcation47H05;47H09

1 Introduction

Nonlinear boundary value problems(NBVPs)involving the p-Laplacian operator arise from many physical phenomena,such as reaction-difusion problems,petroleum extraction,fowthrough porous media and non-Newtonian fuids.Thus,it is a hot topic to study such problems and their generalizations by using diferent methods.Employing theories of the perturbations on ranges of nonlinear operators to discuss the existence of solutions of NBVPs is one of the important methods,related work can be found in[1–7].In 2010,Wei,Agarwal and Wong [8]studied the following nonlinear parabolic boundary value problem involving the generalized p-Laplacian

where 0≤C(x,t)∈Lp(?×(0,T)),ε is a non-negative constant and ? denotes the exterior normal derivative of Γ.Based on the properties of the ranges for pseudo-monotone operators and maximal monotone operators presented in[9],it is shown that(1.1)has solutions Lp(0,T;W1,p(?)),where 2≤p<+∞.

Recently,Wei,Agarwal and Wong[10]studied the following elliptic p-Laplacian-like equation with mixed boundary conditions

By using the perturbation results on the ranges for m-accretive mappings presented in[1], it is shown that(1.2)has solutions in Lp(?)under some conditions,where

In 2012,Wei,Agarwal and Wong[11]studied the following integro-diferential equation with generalized p-Laplacian operator

By using some results on the ranges for bounded pseudo-monotone operator and maximal monotone operator presented in[3,12,13],they obtain that(1.3)has solutions in Lp(0,T; W1,p(?))for 1

In this paper,motivated by the previous work,we shall consider the following parabolic p-Laplacian-like problem

In(1.4),α is the subdiferential of j,i.e.,α=?j,where j:R→R is a proper,convex and lower-semi continuous function,βxis the subdiferential of ?x,i.e.,βx≡ ??x,where?x=?(x,·):R→R is a proper,convex and lower-semicontinuous function.More details of (1.4)will be presented in Section 2.We shall discuss the existence and uniqueness of non-trivial solution of(1.4)in L2(0,T;L2(?)).

The main contributions of this paper lie in three aspects:(i)one of the main partin the previous work is replaced by(ii)an item?u is considered in g(x,u,?u);(iii)the discussion is undertaken in L2(0,T;L2(?)),which does not change while p is varying fromto+∞for N≥1.

We,now,present some preliminaries.

Let X be a real Banach space with a strictly convex dual space X?.We shall use(·,·) to denote the generalized duality pairing between X and X?.We shall use“→ ”and“wlim”to denote strong and weak convergence,respectively.Let“X?→Y”denote the space X embedded continuously in space Y.For any subset G of X,we denote by intG its interior andits closure,respectively.A mapping T:X→X?is said to be hemi-continuous on X(see [12,13])if=Tx for any x,y∈X.

A function Φ is called a proper convex function on X(see[12,13])if Φ is defned from X to(?∞,+∞],not identically+∞,such that Φ((1?λ)x+λy)≤(1?λ)Φ(x)+λΦ(y),whenever x,y∈X and 0≤λ≤1.

Given a proper convex function Φ on X and a point x∈X,we denote by?Φ(x)the set of all x?∈X?such that Φ(x)≤Φ(y)+(x?y,x?),for every y∈X.Such element x?is called the subgradient of Φ at x,and?Φ(x)is called the subdiferential of Φ at x(see[12]).

Let J denote the normalized duality mapping from X into 2X?,which is defned by

If X is reduced to the Hilbert space,then J is the identity mapping.

A multi-valued mapping A:X→2Xis said to be accretive(see[12])if(v1?v2,J(u1?u2))≥0 for any ui∈D(A)and vi∈Aui,i=1,2.The accretive mapping A is said to be m-accretive if R(I+λA)=X for some λ>0.

Proposition 1.1(see[13]) If Φ :X → (?∞,+∞]is a proper convex and lowersemicontinuous function,then?Φ is maximal monotone from X to X?.

Proposition 1.2(see[13]) If A:X → 2X?is a everywhere defned,monotone and hemi-continuous mapping,then A is maximal monotone.If,moreover,A is coercive,thenR(A)=X?.

Proposition 1.3(see[13]) If A1and A2are two maximal monotone operators in X such that(intD(A1))∩D(A2)6=?,then A1+A2is maximal monotone.

Proposition 1.4(see[14]) Let ? be a bounded conical domain in RN.If mp>N,then Wm,p(?)?→CB(?);if 01,then for 1≤q<+∞,then Wm,p(?)?→Lq(?).

2 Main Results

In this paper,unless otherwise stated,we shall assume that N≥1 and m≥0.If p≥2, then m+s+1=p;if

In(1.4),? is a bounded conical domain of a Euclidean space RNwith its boundary Γ∈C1(see[4]),T is a positive constant,ε is a non-negative constant,0≤C(x,t)∈Lmax{p,p′}(0,T; Lmax{p,p′}(?))and ? denotes the exterior normal derivative of Γ.We shall assume that Green’s Formula is available.

Suppose that α≡?j,where j:R→ R is a proper,convex and lower-semi continuous function.βx≡??x,where ?x=?(x,·):R→R is a proper,convex and lower-semi continuous function,for each x∈Γ,0∈βx(0)and for each t∈R,the function x∈?！?I+λβx)?1(t)∈R is measurable for λ>0.Suppose g:?×RN+1→R is a given function satisfying the following conditions

(a)Carath′eodory’s conditions

(b)Growth condition

where(s1,s2,···,sN+1)∈RN+1,h(x)∈L2(?)and l is a positive constant.

(c)Monotone condition

g is monotone with respect to r1,i.e.,

for all x∈? and(s1,···,sN+1),(t1,···,tN+1)∈RN+1.

Now,we present our discussion in the sequel.

Lemma 2.1For p≥2,defne the mapping B:Lp(0,T;W1,p(?))→Lp′(0,T;(W1,p(?))?) by

for any u,w∈Lp(0,T;W1,p(?)).Then,B is strictly monotone,pseudo-monotone and coercive (here,h·,·i and|·|denote the Euclidean inner-product and Euclidean norm in RN,respectively).

Step 1B is everywhere defned.

Case 1s≥0.For u,w∈Lp(0,T;W1,p(?)),we fnd

which implies that B is everywhere defned.

Case 2s<0.For u,w∈Lp(0,T;W1,p(?)),we have

which implies that B is everywhere defned.

Step 2B is strictly monotone.

For u,v∈Lp(0,T;W1,p(?)),we have

Step 3B is hemi-continuous.

It sufces to show that for any u,v,w∈Lp(0,T;W1,p(?))and t∈[0,1],(w,B(u+tv)?Bu)→0 as t→0.Since

by Lebesque’s dominated convergence theorem,we fnd

Hence,B is hemi-continuous.

Step 4B is coercive.

Case 1s≥0.For u∈Lp(0,T;W1,p(?)),letwe fnd

which implies that B is coercive.

Case 2s<0.For u∈Lp(0,T;W1,p(?)),let kukLp(0,T;W1,p(?))→+∞,we fnd

which implies that B is coercive.

Lemma 2.2Fordefne:Lp′(0,T;W1,p(?))→Lp(0,T;(W1,p(?))?)by

Step 1is everywhere defned.

Case 1s≥0.For u,w∈Lp′(0,T;W1,p(?)),we fnd

In view of Proposition 1.4,W1,p(?)?→Lp′(?)?→Lp(?).Then

where k′is a positive constant,which implies thatbB is everywhere defned.

Case 2s<0.For u,w∈Lp′(0,T;W1,p(?)),we have

Similar to the discussions of Steps 2 and 3 in Lemma 2.1,we know thatbB is strictly monotone and hemi-continuous.

Step 2is coercive.

Case 1

Case 2s<0.Then

Lemma 2.3(see[11]) (i)For p≥2,defne the function Φ:Lp(0,T;W1,p(?))→R by

Then Φ is proper,convex and lower-semi continuous on Lp(0,T;W1,p(?)).

Therefore,Proposition 1.1 implies that?Φ:Lp(0,T;W1,p(?))→ Lp′(0,T;(W1,p(?))?), the subdiferential of Φ,is maximal monotone.

Lemma 2.4(see[11]) (i)For p≥2,if w(x,t)∈?Φ(u),then w(x,t)=βx(u)a.e.on ?！?0,T).

Defnition 2.5Defne a mapping A:L2(0,T;L2(?))→ 2L2(0,T;L2(?))in the following way:

(i)if p≥2,

for u∈D(A)={u∈L2(0,T;L2(?))|there exists a w(x)∈L2(0,T;L2(?))such that w(x)∈Bu+?Φ(u)};

for u∈D(A)={u∈L2(0,T;L2(?))|there exists a w(x)∈L2(0,T;L2(?))such that w(x)∈bBu+?Φ(u)}.

Lemma 2.6The mapping A:L2(0,T;L2(?))→L2(0,T;L2(?))defned in Defnition 2.5 is maximal monotone.

ProofFrom Lemmas 2.1–2.3,we can easily get the result that A is monotone.

Next,we shall show that R(I+A)=L2(0,T;L2(?)),which ensures that A is maximal monotone.

Case 1p≥2,then we defne F:Lp(0,T;W1,p(?))→Lp′(0,T;(W1,p(?))?)by

where(·,·)L2(0,T;L2(?))denotes the inner-product of L2(0,T;L2(?)).Then F is everywhere defned,monotone and hemi-continuous,which implies that F is maximal monotone in view of Proposition 1.2.Combining with the facts of Propositions 1.1–1.3 and Lemmas 2.1 and 2.3,we have R(B+F+?Φ)=Lp′(0,T;(W1,p(?))?).

Then for f∈L2(0,T;L2(?))?Lp′(0,T;(W1,p(?))?),there exists u∈Lp(0,T;W1,p(?))?L2(0,T;L2(?))such that

which implies that R(I+A)=L2(0,T;L2(?)).

Case 2

by

which implies that R(I+A)=L2(0,T;L2(?)). ?

Remark 2.7Since L2(0,T;L2(?))is a Hilbert space,Lemma 2.6 presents not only an example of maximal monotone operator but also an m-accretive mapping related to parabolic equation.

Defnition 2.8Defne a mapping H:L2(0,T;L2(?))→2L2(0,T;L2(?))by

Lemma 2.9The mapping H:L2(0,T;L2(?))→L2(0,T;L2(?))defned in Defnition 2.8 is maximal monotone.

ProofStep 1H is everywhere defned.

From condition(b)of g,we know that for u(x,t),v(x,t)∈L2(0,T;L2(?)),

This implies that H is everywhere defned.

Step 2H is hemi-continuous.

Since g satisfes condition(a),we have for any w(x,t)∈L2(0,T;L2(?)),

as t→0,which implies that H is hemi-continuous.

Step 3H is monotone.

In view of condition(c)of g,we have

which implies that H is monotone.

Thus Proposition 1.2 implies that H is maximal monotone.

Lemma 2.10Defne S:D(S)={u(x,t)∈H1(0,T;L2(?)):u(x,0)=u(x,T)∈ L1(0,T;?)}?L2(0,T;L2(?))→(?∞,+∞]by

Then the mapping S is proper,convex and lower-semi continuous.

ProofWe can easily know that S is proper and convex since j is proper and convex. Next,we shall show that S is lower-semi continuous on H1(0,T;L2(?)).

For this,let{un}be such that un→u in H1(0,T;L2(?))as n→∞.Then there exists s subsequence of{un},which is still denoted by{un}such thata.e.(x,t)∈?×(0,T).Since j is lower-semicontinuous,thena.e.on ?×(0,T). Using Fatou’s lemma,we have

Lemma 2.11Let S be the same as that in Lemma 2.10.Similar to Lemma 2.4,we have

Remark 2.12are diferent,S is constructed and Lemmas 2.10 and 2.11 are new results compared to the existing work and will play an important role in the later discussion.

Theorem 2.13For f(x,t)∈L2(0,T;L2(?)),the nonlinear parabolic equation(1.4)has a unique solution u(x,t)in L2(0,T;L2(?)),i.e.,

(b)?h?,(C(x,t)+|?u|2)s

2|?u|m?1?ui∈βx(u(x,t)),a.e.(x,t)∈Γ×(0,T);

(c)u(x,0)=u(x,T),x∈?.

ProofWe split our proof into two steps.

Step 1There exists unique u(x,t)∈L2(0,T;L2(?))which satisfes f=?S(u)+Hu+Au.

From Lemmas 2.6,2.9 and 2.10 and Propositions 1.1 and 1.3,we know that there exists u(x,t)which satisfes?S(u(x,t))+Au(x,t)+Hu(x,t)=f(x,t),where f(x,t)∈L2(0,T;L2(?)) is a given function.Next,we shall prove that u(x,t)is unique.

Suppose that u(x,t)and v(x,t)satisfy?S(u)+Au+Hu=f and?S(v)+Av+Hv=f, respectively.Then,0≤(u?v,Au?Av)=?(u?v,(?S+H)u?(?S+H)v)≤0,which ensures that either(u?v,Bu?Bv)=0 or(u?v,bBu?bBv)=0.This implies that u(x,t)=v(x,t), since both B andbB are strictly monotone.

Step 2If u(x,t)∈L2(0,T;L2(?))satisfes f=?S(u)+Hu+Au,then u(x,t)is the solution of(1.4).

which implies that the equation

is true.

By using(2.1)and Green’s Formula,we have for p≥2,

Then(2.2)and(2.3)imply that

From the defnition of S,we can easily obtain u(x,0)=u(x,T)for all x∈?.Combining with(2.1)and(2.4)we see that u is the unique solution of(1.4). ?

Theorem 2.14If we suppose further that 0∈?j(0)and g(x,θ)≡0 for x∈? and θ=(0,0,···,0)∈RN+1,then for 0 6=f∈L2(0,T;L2(?)),equation(1.4)has a unique non-trivial solution in L2(0,T;L2(?)).

ProofFrom Theorem 2.13,we know that for 0 6=f∈L2(0,T;L2(?)),(1.4)has a unique solution u(x,t)∈L2(0,T;L2(?)).Next,we shall show that u(x,t)6=0.

If,on the contrary,u(x,t)=0,then from(a)in Theorem 2.13,we know that f(x,t)=0, which makes a contradiction! ?

Remark 2.15If,in(1.4),the function α≡I(the identity mapping),then it reduces to the following one

If,in(2.5),m=1 and s=p?2,then it becomes to the

If,moreover,s=0 and m=p?1,then(2.6)becomes to the case of parabolic p-Laplacian problems.

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?Received April 13,2015;revised May 1,2016.The frst author is supported by the National Natural Science Foundation of China(11071053),Natural Science Foundation of Hebei Province(A2014207010),Key Project of Science and Research of Hebei Educational Department(ZD2016024)and Key Project of Science and Research of Hebei University of Economics and Business(2015KYZ03).