3 Proof of the main theorem

L ₁ -maximal regularity for quasilinear second order differential equation with damped term

Kordan Ospanov

L. N. Gumilyov Eurasian National University, 2 Satpaev Street, Astana, 010008, Kazakhstan Received 7 March 2015, appeared 13 July 2015

Communicated by Michal Feˇckan

Abstract. We investigated a quasilinear second order equation with damped term on the real axis. We gave some suitable conditions for existence of the L₁-maximal regular solutions of this equation.

Keywords: differential equation, nonlinear damped term, separation of operator, fixed point theorem.

2010 Mathematics Subject Classification: 34A34.

1 Introduction and statement of main result

LetR:= (−_∞,+_∞)_,L₁:= L₁(_R)_andk · k₁ be the norm ofL₁. We consider the equation Ly:= −y⁰⁰+r(x,y)y⁰+q(x,y)y= f(x), x∈_R, (1.1) where ris continuously differentiable andqis a continuous function, f ∈ L₁. This is a useful equation in mathematical physics (see [5,26]).

ByC₀^(k)(_R)(k=1, 2, . . .) we denote the set ofktimes continuously differentiable functions with compact support. Let C_loc^(j)(_R) =y :ψy∈C⁽₀^j)(_R),∀ψ∈C⁽₀^∞)(_R) (j=1, 2).

Definition 1.1. Lety ∈L₁, if there is a sequence {y_n}^∞_n=1 ⊂C_loc⁽²⁾(_R)such that

nlim→_∞kψ(y_n−y)k₁=0 and lim

n→_∞kψ(Ly_n− f)k₁ =0,

∀ψ∈C₀^(∞)(_R)_{. Then}yis called a solution of (1.1).

The purpose of this work is to find some conditions forr andqsuch that for every f ∈L₁, the equation (1.1) has a solutionywhich satisfies

ky⁰⁰k₁+kr(·,y)y⁰k₁+kq(·,y)yk₁ <_∞.

BEmail: ospanov_kn@enu.kz

The separability of differential operators introduced by Everitt and Giertz in [7, 8] plays an important role in the study of second order differential equations. Recall that the Sturm–

Liouville operator

eLy:=−y⁰⁰+q₁(x)y,

acting inL2(_R)is separable, if there is a constant c>0 such that

k −y⁰⁰k₂+kq₁(·)yk₂ ≤c(keLyk₂+kyk₂), ∀ y∈ D(eL).

Everitt and Giertz [7,8] proved that if q₁ and its derivatives satisfy some conditions, then eL is separable in L₂(_R). In the case q₁ is not differentiable function, the separability of eL in L2(_R) was discussed in [3, 23]. In [9], Everitt, Giertz and Weidmann give an example of non-separable Sturm–Liouville operator in L₂(_R) with strongly oscillating and infinitely smooth coefficientq₁. The separability of linear partial differential operators was studied in [4,15,17,21,24]. Some sufficient conditions of separability of operators on Riemann manifolds are obtained in [1,2,12,13].

The separability is also an important tool when dealing with quasilinear equations. In [16], Muratbekov and Otelbaev used the separability to discuss the solvability of the nonlinear equation

−y⁰⁰+q₀(x,y)y= f(x), (1.2) where f ∈ L₂(_R). Grinshpun and Otelbaev showed that the solvability of the equation (1.2) in L₁ implies q₀ ≥ 1 (see [6]). This method is useful for the multidimensional (Schrödinger) equation−_∆u+q(x,u)u= F(x)_, x ∈_Rⁿ(see [17,20] for details).

In general, the expression (1.1) can be converted neither to (1.2) nor to the form

− p₂(x,y)y⁰0

+q₂(x,y)y= f(x)_.

In [22], we considered the equation−y⁰⁰+r(x,y)y⁰ = f(x), f ∈ L₂(_R), and found some conditions forr such that this equation is solvable. In the present paper, we discuss the more general equation (1.1), in the case f ∈ L₁. Under weaker conditions on r than in [22] the existence and regularity of solutions of (1.1) are established.

Schauder’s fixed-point theorem is used to prove our main result (see [10]).

Let gandh be some functions onRand let α_g,h(t) =

Z _t

0

|g(η)|dηess sup

ξ∈(t,+_∞)

|h(ξ)|⁻¹, t>0, β_g,h(τ) =

Z ₀

τ

|g(η)|dη ess sup

ξ∈(−_∞,τ)

|h(ξ)|⁻¹, τ<0, γ_g,h=max

sup

t>0

α_g,h(t), sup

τ<0

β_g,h(τ)

. The main result of this paper is the following.

Theorem 1.2. Let r be a continuously differentiable function and q a continuous function satisfying r≥δ₁

p1+x² (δ₁ >0) (1.3)

and

sup

c0∈_R

γ_{q(·,c0),r(·,c0)}< _∞. (1.4)

Then for any f ∈ L₁, the equation(1.1)has a solution y such that

ky⁰⁰k₁+kr(·,y)_y⁰k₁+kq(·,y)_yk₁ <_{∞. (1.5)} Example 1.3. Letr =10+x¹⁰+5y⁴,q=x³+cos⁴x+2y. Thenr andqsatisfy the conditions of Theorem1.2.

2 Auxiliary statements

By Muckenhoupt’s theorem (Theorem 2 in [14]), we obtain the following lemma.

Lemma 2.1. Let g and h be continuous functions onRsuch thatγ_g,h< _{∞. Then}

Z

R|g(x)y(x)|dx ≤γ_g,h Z

R|h(x)y⁰(x)|dx, ∀y∈C₀⁽¹⁾(_R). (2.1) Moreover,γ_g,his the smallest constant which satisfies(2.1).

Letrbe a continuously differentiable function and

l₀y =−y⁰⁰+r(x)y⁰, D(l₀) =C₀⁽²⁾(_R). Denote bylthe closure ofl₀ in L₁.

Lemma 2.2. Let r be continuously differentiable and satisfy(1.3). Then l is invertible, R(l) = L₁and ky⁰⁰k₁+kry⁰k₁ ≤3klyk₁, ∀y∈ D(l). (2.2) Proof. We use the method of [25] to prove (2.2). Let y ∈ C⁽₀²⁾(_R) be a real function, and γ> −1. Then by integration by parts, we get

Z

R(ly)y⁰

(y⁰)^2γ/2 dx =

Z

Rr

(y⁰)^2γ/2+1 dx.

By Hölder’s inequality, we have that Z

Rr

(y⁰)^2γ/2+1 dx≤ _Z

R|r^−αly|^pdx _1/p_Z

R|r^α(y⁰)^γ+1|^qdx 1/q

, (2.3)

where 1 < p < _∞, q = p/(p−1). We choose αandγ as follows: (γ+1)q = γ+2, αq= 1.

This implies γ+₂ = p. So from (2.3) it follows that kr^1/py⁰k_p ≤ kr^−1/qlyk_p. Taking the limit p →1, we get

kry⁰k₁≤ klyk₁, y∈C₀⁽²⁾(_R). (2.4) Sinceky⁰⁰k₁≤ klyk₁+kry⁰k₁ ≤2klyk₁, we haveky⁰⁰k₁+kry⁰k₁ ≤3klyk₁,∀y∈C₀⁽²⁾(_R). Since lis a closed operator, the last inequality holds for anyy∈ D(l).

From Lemma2.1, (1.3) and (2.4) follows that

kyk₁≤c₁klyk₁, y∈ D(l). (2.5) So the inversel⁻¹ oflexists.

Next, we show thatR(l) =L₁. Let R(l)6=L₁. Sincelis closed, (2.5) impliesR(l)is closed.

Hence there exists a nonzero elementz0 ∈ ^⊥R(l)such thatl^∗z0 =−z⁰⁰₀−(r(x)z0)⁰ =0, where l^∗ is adjoint operator ofl. Then

z₀exp

_{Z x}

a r(η)dη ⁰

=c₂exp _{Z x}

a r(η)dη

. Letc₂6=0. Without loss of generality, we can assume that c₂=−1. Then

z₀exp

_{Z x}

a r(η)dη ⁰

<0,

i.e. z₀(x) is a monotonically decreasing function, moreoverz₀(x−k) > exp(k)z₀(x), x ∈ _R, k=1, 2, . . ., which implies thatz₀ ∈/ L_∞(_R)_.

Letc₂=0. Then

z0=c3exp

−

Z _x

a r(η)dη

, c3 6=0.

Thereforez0 ∈/ L_∞(_R). We obtain a contradiction, hence R(l) =L₁. We consider the following linear equation

ly:=−y⁰⁰+r₁(x)y⁰+q₁(x)y= f(x), x∈_R. (2.6) The functiony ∈ L₁ is called a solution of (2.6), if there exists a sequence{y_n}^∞_n=1 ⊂ C₀⁽²⁾(_R) such thatky_n−yk₁→0 andkly_n− fk₁ →0 asn→_∞.

Lemma 2.3. Let r₁ be a continuously differentiable function such that r₁ ≥δ₁

√1+x².Assume q₁is a continuous function andγq₁,r₁ <∞. Then for every f ∈ L₁, the equation(2.6)has a unique solution y such that

ky⁰⁰k₁+kr₁y⁰k₁+kq₁yk₁≤ c₄kfk₁, (2.7) where c₄depends only onγq₁,r₁.

Proof. Letx =at(a>0), then (2.6) becomes that

−y˜⁰⁰+a⁻¹r˜₁(t)y˜⁰+a⁻²q˜₁(t)y˜= f^˜, (2.8) where ˜y(t) =y(at), ˜r₁(t) =r₁(at), ˜q₁(t) =q₁(at), ˜f(t) = a⁻²f(at). Letl0ay˜ = −y˜⁰⁰_tt+a⁻¹r˜₁y˜⁰,

˜

y∈ C⁽₀²⁾(_R). Byl_a we denote the closure in L₁of l_0a. Since a⁻¹r˜₁(t)satisfies the conditions of Lemma2.2, it follows that the operatorl_a is continuously invertible and

ky˜⁰⁰k₁+ka⁻¹r˜₁y˜⁰k₁ ≤3klay˜k₁, ∀y˜∈ D(la). (2.9) Leta=_{4 1}+γ_q˜1,˜r1

. By2.1, we obtain that ka⁻²q˜₁y˜k₁ ≤ ^γq˜1,˜r1

a² kr˜₁y˜⁰k₁≤ ³

4kl_ay˜k₁, ∀y˜ ∈ D(l_a). (2.10) Hence, by a well-known theorem (see [11, Chapter 4, Theorem 1.16]), we find that the operator la+a⁻²q˜₁(t)Ecorresponding to (2.8) is invertible andR la+a⁻²q˜₁E

=L₁. Let ˜ybe a solution of the equation (2.8), by (2.9) and (2.10), we obtain that

ky˜⁰⁰k₁+ka⁻¹r˜₁y˜⁰k₁+ka⁻²q˜₁y˜k₁ ≤₄kl_ay˜k_{1. (2.11)}

From (2.10) it follows that

ka⁻²q˜₁y˜k₁≤₃k l_a+a⁻²q˜₁E

˜ yk_1. So

klay˜k₁≤ k la+a⁻²q˜₁E

y˜k₁+ka⁻²q˜₁y˜k₁ ≤4k la+a⁻²q˜₁E

y˜k₁. (2.12) The inequalities (2.11) and (2.12) imply that for the solution ˜yof (2.8), the following inequality holds:

ky˜⁰⁰k₁+ka⁻¹r˜₁y˜⁰k₁+ka⁻²q˜₁y˜k₁≤16kf^˜k₁. Takingt =a⁻¹x, we to obtain (2.7).

Remark 2.4. Letr₁(x)be continuously differentiable. Assumeq₁(x)is a continuous function.

L will denote the closure in L₁ of the operator L₀y := −y⁰⁰+r₁y⁰+q₁y, D(L₀) = C⁽₀²⁾(_R). If there is a constantc5>0 such thatk −y⁰⁰k₁+kr₁(·)y⁰k₁+kq₁(·)yk₁≤c5(kLyk₁+kyk₁),∀ y∈ D(L), thenL is called separable inL₁.

If the conditions of Lemma2.3 hold, then the operator Lis separable inL1.

3 Proof of the main theorem

Let C(_R) be the space of bounded continuous functions on R with the norm kyk_C(R) = sup_x∈R|y(x)|. Letεand Abe positive numbers. Set

S_A= (

z∈C(_R): sup

x∈_R

|z(x)| ≤ A )

.

Letv∈S_A. L_v,ε denote the closure inL₁of the following linear differential expression L_0,v,εy:=−y⁰⁰+r(x,v(x)) +ε 1+x²

y⁰+q(x,v(x))y, ∀y∈ C₀⁽²⁾(_R). We consider the equation

L_v,εy= f(x). (3.1)

˜

r_1,ε,v(x) := r(x, v(x)) +ε 1+x²

and ˜qv(x) := q(x,v(x)) satisfy all of the conditions of Lemma 2.3. Indeed, by (1.3), ˜r_1,ε,v(x) ≥ δ₁

√

1+x², δ₁ > 0. Hence γ_1,˜r1,ε,v < ∞. From (1.4) it follows thatγ_{q˜v(x),˜r1,ε,v(x)} ≤C₁sup_t∈Rγ_{q(x,t),r(x,t)} <∞. Therefore, for any f ∈ L₁, the equation (3.1) has a unique solution yand

ky⁰⁰k₁+

r(·, v(·)) +ε(1+x²)y⁰

1 +kq(·, v(·))yk₁ ≤C2kfk₁, (3.2) whereC₂ does not depend on A. By2.1, we have that

kyk₁ ≤C3kr˜_1,ε,vy⁰k₁,

p1+x²y

1 ≤C₄

1+x² y⁰

1. (3.3)

By using (3.2), (3.3) and Theorem 1 given in Chapter 3 of [18], we obtain that kyk_W :=ky⁰⁰k₁+

r(·, v(·)) + (1+x²)y⁰ 1+

h|q(·, v(·))|+^p1+x²i y

1

+sup

x∈_R

(1+x²)^3/8y(x)

≤C₅kfk₁, y∈ D(L_v,ε), (3.4)

whereC₅ also does not depend on A.

Let A = C5kfk₁+1. SetPε(v) = L⁻_v,ε¹f, wherev ∈ SA, ε >0, f ∈ L₁ and L⁻_v,ε¹ is inverse to L_v,ε. According to (3.4), the operator Pε maps S_A into itself. Moreover, the operator Pε maps S_Ato the set

Q_A:= {y:kyk_W ≤ C5kfk₁}.

Q_A is compact inC(_R). Indeed, letγ > 0, then by (3.4) there exists l ∈ _Nsuch that for any z∈ Q_A

kz⁰⁰k_L

1(_R\[−l,l])+

r(·_, v(·)) + (₁+x²)z⁰

L1(_R\[−l,l])

+

h|q(·, v(·))|+^p1+x²i z

L1(_R\[−l,l]) <γ/2, (3.5) and

sup

x:|x|≥l

C₅kfk₁+1

(1+x²)^3/8 <γ/2. (3.6)

Let ϕ_l ∈C₀^(∞)(−l−1, l+1) (l=1, 2, . . .) such that ϕ_l(x) =1 for x∈[−l, l],ϕ_l(x) =0 for x ∈/ [−l−1, l+1]and 0 ≤ ϕ_l ≤1. We denote T_l = {ϕ_lz:z∈ QA}. By (3.5) and (3.6),T_l is a γ-net ofQ_A. On the other hand,T_l is a subset of the Sobolev space

◦

W²₁(−l−1, l+₁) =θ∈W₁²(−l−1, l+₁)_:θ(x) =_{0, as}|x| ≥l+_{1 .}

Notice that the embedding of W^{◦ 2}₁(−l −1,l+1) in C₀[−l−1,l+1] is compact, where C₀[−l−1,l+1] = {η∈ C[−l−1, l+1]:η(x) =0, as|x| ≥l+1} (see [19, 27]). So T_l is a compactγ-net ofQ_A. By Hausdorff’s theorem (see [10, Chapter 1]),Q_A is compact inC(_R).

Next, we show that the operator Pε is continuous on S_A. Let{v_n}^∞_n=1 ⊂ S_Abe a sequence such that sup_x∈R|v_n(x)−v(x)| →0 asn→+_∞. Ify_n (n=1, 2, . . .) andysatisfy

L_v,εy= f, L_vn,εy_n= f. (3.7)

ThenPε(vn)−Pε(v) =yn−y. So it suffices to show that sup_x∈R|yn(x)−y(x)| →0 asn→_∞.

By (3.7), we deduce that

y−yn =_L⁻_v,ε¹[r(_x,vn(_x))−r(_x,v(_x))]_y⁰_n+ [_q(_x,vn(_x))−q(_x,v(_x))]_{yn . (3.8)} Since functions v and v_n (n = 1, 2, . . .) are continuous, we see that r(x,v_n(x))−r(x,v(x)) and q(x,v_n(x))−q(x,v(x))are continuous functions. Therefore, from (3.8) it follows that

kyn−yk_L1(−a,a)≤c max

x∈[−a,a]

r(x,vn(x))−r(x,v(x))|,|q(x,vn(x))−q(x,v(x))|

×^hky⁰_nk_L1(−a,a)+kynk_L1(−a,a)ⁱ→0

(3.9)

as n → ∞, for everya > 0. On the other hand, by (3.4), we have {y_n}⁺_n=^∞₁ ⊂ Q_A, ky_nk_W ≤ A(n = 1, 2, . . .), y ∈ Q_A, kyk_W ≤ A. Since the set Q_A is compact in C(_R), without loss of generality, we can assume that the sequence{yn}⁺_n=^∞₁converges to somez∈ C(_R). By (3.4)

|xlim|→_∞y(x) =0, lim

|x|→_∞z(x) =0. (3.10)

Since the operatorL⁻_v,ε¹ is closed, by (3.9) and (3.10), we obtain thatz =y. ThusP_ε is continu- ous.

SoP_ε is a completely continuous operator inC(_R)and it maps the ball S_A into itself. By Schauder’s theorem (see [10, Chapter XVI]), Pε has a fixed pointy inSA, i.e.Pε(y) = y. And ysatisfies the equality

−y⁰⁰+r(x, y) +ε(1+x²)y⁰+q(x,y)y= f(x). By Lemma2.3, we obtain that

ky⁰⁰k₁+kr(·, y) +ε(1+x²)y⁰k₁+kq(·, y)y⁰k₁ ≤C₅kfk₁. Now, let

ε_j ^∞_j=1 be a sequence of positive numbers such that lim_j→_∞ε_j = 0. Recall that Pε_j(v) =L⁻_v,ε¹_jf. Ify_j ∈S_Ais the fixed point of the operatorPε_j, then

−y⁰⁰_j +r(x, y_j) +ε_j(1+x²)y⁰_j+q(x,y_j)y_j = f(x). Then according to Lemma2.3, we have

y⁰⁰_j

1+

r ·, y_j(·)+ε_j 1+x² y⁰_j

1 +q ·, y_j(·)y_j

1 ≤C5kfk₁. (3.11) Let (a,b) be an arbitrary finite interval. It is known that the spaceW₁²(a, b)is compactly embedded to L₁(a, b). Therefore, by virtue of (3.11), we can select a subsequence

˜

y_j ^∞_j=1 of y_j ^∞_j=1⊂W₁²(a, b)such thatky˜_j−yk_L1(a,b)→0 asj→∞. By Definition1.1,yis a solution of equation (1.1). By Lemma2.3, we obtain that forythe estimate (1.5) holds.

Remark 3.1. The condition (1.3) is natural. If (1.3) does not hold, from Lemma2.1 it follows that the domain D(L)ofLis not included inL₁.

Acknowledgements

This work is supported by project 5132/GF4 of Science Committee of Ministry of Education and Science of Republic of Kazakhstan.

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