We study properties of the Lasota partial differential equation in two different spaces: \(V_{\alpha}\) (Hölder continuous functions) and \(L^p\). The aim of this paper is to generalize the results of [Z. Brzeźniak, A. L. Dawidowicz, On the periodic solution to the von Foerster-Lasota equation, to appear in Semigroup Forum]

Anna Poskrobko

Antoni Leon Dawidowicz

Dawidowicz, Antoni

Poskrobko, Anna

Jagiellonian Univeristy Repository

Opuscula Mathematica • Vol. 28 • No. 4 • 2008Professor Andrzej Lasota in MemoriamAntoni Leon Dawidowicz, Anna PoskrobkoON PERIODIC AND STABLE SOLUTIONSOF THE LASOTA EQUATIONIN DIFFERENT PHASE SPACESAbstract. We study properties of the Lasota partial differential equation in two differentspaces: Vα (Hölder continuous functions) and Lp. The aim of this paper is to generalize theresults of [1].Keywords: partial differential equations, periodic solutions, stable solutions.Mathematics Subject Classification: 35B10, 35B35, 37C75, 47D06.1. INTRODUCTIONWe consider the partial differential equation∂u∂t+ x∂u∂x= λ(x)u, t ≥ 0, 0 ≤ x ≤ 1 (1.1)with the initial conditionu(0, x) = v(x), 0 ≤ x ≤ 1, (1.2)where v belongs to some normed vector space V of functions defined on [0, 1] andλ : [0, 1] → R is a given continuous function. Let a semidynamical systemTt : V → Vbe given by the formula(Ttv)(x) = u(t, x),where u is the solution of (1.1), (1.2). It is clear that this unique solution is given bythe formula(Ttv)(x) = u(t, x) = eg(x)e−g(xe−t)v(xe−t), x ∈ [0, 1], (1.3)453454 Antoni Leon Dawidowicz, Anna Poskrobkowhereg(x) = −∫ 1xλ(s)sdswith the condition ∫ 10λ(s)sds = ∞. (1.4)We wish to investigate some properties of system (1.3): periodic solutions, strong andexponential stability.Definition 1.1. A function v0 ∈ V is a periodic point of the semigroup (Tt)t≥0, witha period t0 ≥ 0 iff Tt0v0 = v0. A number t0 > 0, is called a principal period of aperiodic point v0 iff the set of all periods of v0 is equal to Nt0.Definition 1.2. The semigroup (Tt)t≥0 is strongly stable in V iff for every v ∈ V ,limt→∞Ttv = 0 in V.Definition 1.3. The semigroup (Tt)t≥0 is exponentially stable iff there exist D < ∞and ω > 0 such that‖Tt‖ ≤ De−ωt, for t ≥ 0.The problem of the chaotic behaviour of a partial differential equation was consid-ered by Lasota [5], Rudnicki [8], Łoskot [7] and Szarek [6]. In the papers [1–4] therewere described properties of the partial differential equation, analogical to (1.1), butwith a constant function λ:∂u∂t+ x∂u∂x= γu, t ≥ 0, 0 ≤ x ≤ 1 (1.5)and with the initial conditionu(0, x) = v(x), 0 ≤ x ≤ 1. (1.6)This work has been intended as an attempt at generalizing the results of [1]. In [1]there was described the chaotic and stability behaviour of the suitable semidynamicalsystem(T̃tv)(x) = ũ(t, x) = eγtv(xe−t), x ∈ [0, 1] (1.7)in different phase spaces V . All properties depended on the value γ. We are interestedin finding a connection between this two equations. It is easy to check that if u andũ are the solutions of equation (1.1) and (1.5), respectively, thenũ(t, x) = κ(x)u(t, x), (1.8)whereκ(x) = eR x0λ(0)−λ(s)s ds and γ = λ(0). (1.9)Hence the diagramOn periodic and stable solutions of the Lasota equation 455VTt−−−−→ Vmκy ymκV −−−−→eTt VThis substitution will be a useful tool. It will be used in the proofs of theoremson chaos and stability of system (1.3) in the spaces Vα and Lp.2. PROPERTIES OF THE DYNAMICAL SYSTEM (Tt)t≥0 IN THE SPACE VαLet v be a continous function on [0, 1] such that v(0) = 0. For every interval A ⊂ [0, 1]and for every α ∈ (0, 1], defineHA,α(v) = supx,y∈A,x 6=y|v(x)− v(y)||x− y|α.A function v for which HA,α(v) < ∞ is called a Hölder continuous on the interval Awith exponent α. WriteHα = H[0,1],α.Definition 2.1. Denote by Vα the space of all Hölder continuous functions v on [0, 1]with exponent α, vanishing at zero and satisfying the following conditionlimx→0H[0,x],α(v) = 0.Certain properties of system (1.7) in the space Vα have been estabilished. Forγ > α, there exist periodic solutions of problem (1.5) and the set of all periodicpoints is dense in Vα. Strong and exponential stability take place, provided thatγ ≤ α and γ < α, respectively (see [1] for more details).Theorem 2.2. Let|λ(0)− λ(x)| ≤ Cx, C > 0, x ∈ [0, 1] (2.1)hold. Then the function u ∈ Vα if and only if ũ ∈ Vα.Proof. The assumption u ∈ Vα means that u is a Hölder continuous function withexponent α, vanishing at zero and lima→0 H[0,a],α(u) = 0. This givesH[0,a],α(ũ) = supx,y∈[0,a],x6=y|ũ(t, x)− ũ(t, y)||x− y|α== supx,y∈[0,a],x 6=y|κ(x)u(t, x)− κ(y)u(t, y)||x− y|α≤≤ supx∈[0,a]|κ(x)| · supx,y∈[0,a],x 6=y|u(t, x)− u(t, y)||x− y|α++ supy∈[0,a]|u(t, y)| · supx,y∈[0,a],x 6=y|κ(x)− κ(y)||x− y|α.456 Antoni Leon Dawidowicz, Anna PoskrobkoUsing (1.9) and (2.1), we obtain|κ(x)| ≤ eR x0|λ(0)−λ(s)|s ds ≤ eR x0 Cds = eCx.It remains to show that the function κ is Hölder continous. If κ1α apears the Lipschitzfunction, it will mean that κ is Hölder continuous with exponent α. For x ∈ [0, a]|(κ 1α (x))′| =∣∣∣∣e 1α R x0 λ(0)−λ(s)s ds · 1α · λ(0)− λ(x)x∣∣∣∣ ≤ ∣∣∣∣e 1α R x0 Cds · Cα∣∣∣∣ ==∣∣∣∣eCxα · Cα∣∣∣∣ ≤ eCaα · Cα .Summarizing,H[0,a],α(ũ) ≤ eCa(H[0,a],α(u) + supy∈[0,a]|u(t, y)| ·(Cα)α).Therefore, lima→0 H[0,a],α(ũ) = 0 and finally ũ ∈ Vα. The rest of the proof runsas before. We can draw the same conclusion for the funcition u, assuming thatũ ∈ Vα.Assumption (2.1) will be needed throughout this section and the above theoremwill be crucial for next results.Theorem 2.3. If λ(0) > α ∈ (0, 1], then for any t0 there exists such v0 ∈ Vα thatTt0v0 = v0 (2.2)andTtv0 = v0 if and only if t = nt0 for some positive integer n. (2.3)Proof. Let w be an arbitrary Hölder continuous function with the exponent α definedon the interval [e−t0 , 1] and satisfying the following conditions:e−g(e−t0 )w(e−t0) = w(1), (2.4)∀t ∈ (0, t0) e−g(e−t)w(e−t) 6= w(1). (2.5)Consider the following function v on the interval (0, 1] :v(x) = eg(x)e−g(xent0 )w(xent0) for x ∈ [e−(n+1)t0 , e−nt0 ].The function v is defined on the whole interval (0, 1] =⋃∞n=0(e−(n+1)t0 , e−nt0 ]and come into being by squeezing the graph of the function w into each interval(e−(n+1)t0 , e−nt0 ].By the assumption of the continuity of w on [e−t0 , 1], its boundedness follows, i.e.,On periodic and stable solutions of the Lasota equation 457there is M > 0 such that |w(x)| ≤ M for each x ∈ [e−t0 , 1]. By the above, forx ∈ [e−(n+1)t0 , e−nt0 ], the following estimate holds:|v(x)| = eg(x)e−g(xent0 )|w(xent0)| ≤ Meg(x) · supx∈[e−t0 ,1]e−g(x) ≤ M1eg(x),where M1 = M · supx∈[e−t0 ,1] e−g(x). From assumption (1.4), limx→0 eg(x) = 0, sowe deduce that v(0) = 0. We obtain the continuous function v defined on the wholeinterval [0, 1]. Property (2.2) follows from (2.4), while property (2.3) from (2.5).The assumption λ(0) > α yields ṽ ∈ Vα, see [1] for details. Under Theorem 2.2,we see at once that v ∈ Vα.Theorem 2.4. For λ(0) > α, the set of periodic points of (1.1) is dense in Vα.Proof. Let v be an arbitrary function belonging to Vα. Definew(x) = eg(x)(e−g(xent0 )v(xent0)−∞∑k=n+1me−g(xekt0 )), (2.6)where m = e−g(e−t0 )v(e−t0)−v(1) and x ∈ [e−(n+1)t0 , e−nt0 ]. To show the correctnessof this definition, it is sufficient to make the following observationeg(e−(n+1)t0 )(e−g(e−(n+1)t0ent0 )v(e−(n+1)t0ent0)−∞∑k=n+1me−g(e−(n+1)t0ekt0))== eg(e−(n+1)t0 )(v(1) + m−∞∑k=n+1me−g(e−(n+1)t0ekt0))== eg(e−(n+1)t0 )(e−g(e−(n+1)t0e(n+1)t0)v(e−(n+1)t0e(n+1)t0) −−∞∑k=n+2me−g(e−(n+1)t0ekt0)).The function w is continuous and vanishes at 0, which is a consequence of (2.6).Let ε > 0. Since v ∈ Vα and w ∈ Vα, there exists such t0 that H[0,e−t0 ],α(v) < ε4 andH[0,e−t0 ],α(w) < ε4 . From (1.8) we know that w(x) =ew(x)κ(x) , where w̃ is the periodicsolution of (1.5). For λ(0) > α the set of periodic points of (1.5) is dense in Vα (see[1]), so Hα(v − w̃) < ε4 and Hα(w − w̃) <ε4 .ThusHα(v − w) ≤ H[e−t0 ,1],α(v − w) + H[0,e−t0 ],α(v − w) ≤≤ Hα(v − w̃) + Hα(w̃ − w) + H[0,e−t0 ],α(v) + H[0,e−t0 ],α(w) << ε.This completes the proof.458 Antoni Leon Dawidowicz, Anna PoskrobkoTheorem 2.5. If λ(0) ≤ α and v ∈ Vα, thenlimt→∞Hα(Ttv) = 0.Moreover, if λ(0) < α, then the semigroup (Tt)t≥0 is exponentially stable.Proof. Take any v ∈ Vα. Using Theorem 2.2 and proceeding analogously as in itsproof, we computeHα(Ttv) = supx,y∈[0,1],x 6=y|u(t, x)− u(t, y)||x− y|α= supx,y∈[0,1],x 6=y∣∣∣ eu(t,x)κ(x) − eu(t,y)κ(y) ∣∣∣|x− y|α≤≤ supx∈[0,1]∣∣∣∣ 1κ(x)∣∣∣∣ · supx,y∈[0,1],x 6=y|ũ(t, x)− ũ(t, y)||x− y|α++ supy∈[0,1]|ũ(t, y)| · supx,y∈[0,1],x 6=y∣∣∣ 1κ(x) − 1κ(y) ∣∣∣|x− y|α≤≤ eC ·(Hα(Ttṽ) + supy∈[0,1]|ũ(t, y)| ·(Cα)α).We know that Ttṽ → 0 in Vα for every ṽ ∈ Vα. The claim limt→∞Hα(Ttṽ) = 0 isbased on the results of paper [1]. From the same source, we have derived the estimate‖Ttṽ‖ ≤ e(γ−α)t‖ṽ‖.|ũ(t, y)| = |(Ttṽ)(y)| = |(Ttṽ)(y)− (Ttṽ)(0)| ≤ Hα(Ttṽ)yα ≤ Hα(Ttṽ),hence, since λ(0) < α, there follows the exponential stability of the semigroup (Tt)t≥0with D = eC(1 +(Cα)α) and ω = α− λ(0).3. CHAOS AND STABILITY OF THE SYSTEM (Tt)t≥0 IN THE SPACE LpTheorem 3.1. Assume that∃C, q > 0 ∀x ∈ [0, 1] |λ(0)− λ(x)| ≤ Cxq. (3.1)The function u belongs to the space Lp if and only if ũ ∈ Lp.Proof. Using (3.1), substitution (1.8) discussed in Section 1 and assuming that u ∈ Lp,we obtain‖ũ(t, x)‖p =∫ 10|κ(x)u(t, x)|pdx ≤∫ 10epR x0|λ(0)−λ(s)|s ds|u(t, x)|pdx ≤≤∫ 10eCpq xq|u(t, x)|pdx ≤ eCpq∫ 10|u(t, x)|pdx == eCpq ‖u(t, x)‖p < ∞In the same manner, we can establish the inverse implication.On periodic and stable solutions of the Lasota equation 459The above theorem will be significant in obtaining the next one. It enables theresults of [1] to be used and generalized. From now on, we assume (3.1).Theorem 3.2. For λ(0) > − 1p there exists a periodic solution of (1.1).Proof. For any t0, define the following function v :v(x) = eg(x)e−g(xent0 )w(xent0), x ∈ [e−(n+1)t0 , e−nt0 ], (3.2)v(0) = 0,where w is an arbitrary function from the space Lp. We have shown that such functionv is a periodic solution, so it is sufficient to prove that v ∈ Lp. The function v is thesolution of equation (1.1), so we can express it using the function ṽ, the solution of(1.5), v(x) = ev(x)κ(x) . Brzeźniak and one of the authors [1] showed that ṽ is a periodicsolution and belongs to Lp when γ > − 1p . Our assumption and Theorem 3.1 guaranteethe same conclusion for the function v.Theorem 3.3. If λ(0) > − 1p , then the set of periodic points is dense in Lp.Proof. Let w ∈ Lp and ε > 0. Fix t0 such that[∫ e−t00|w(x)|pdx] 1p<ε2andeCq ‖ṽ|| < ε2, C > 0where ṽ is a periodic solution of (1.5).The function v is defined by formula (3.2). The function v belongs to the set ofperiodic points due to Theorem 3.2. It is sufficient to estimate ||v − w||. Sincev(x) = w(x) for x ∈ [e−t0 , 1], it is obvious that ||v − w|| = ||(v − w)1[0,e−t0 ]||, where1[0,e−t0 ] denotes the indicator of the set [0, e−t0 ]. Applying the estimate from Theorem3.1 and substitution (1.8), we can assert that‖v − w‖ ≤ ‖v1[0,e−t0 ]‖+ ‖w1[0,e−t0 ]‖ < ε.Theorem 3.4. If λ(0) ≤ − 1p , and v ∈ Lp thenlimt→∞‖Ttv‖ = 0.Moreover, for λ(0) < − 1p the semigroup (Tt)t≥0 is exponentially stable on Lp.460 Antoni Leon Dawidowicz, Anna PoskrobkoProof. Let v ∈ Lp be an arbitrary function.‖Ttv‖p =∫ 10|u(t, x)|pdx =∫ 10∣∣∣∣ ũ(t, x)κ(x)∣∣∣∣p dx ==∫ 10∣∣∣∣ 1κ(x) (Ttṽ)(x)∣∣∣∣p dx ≤ eCpq ‖Ttṽ‖p,where C > 0. We know that ‖Ttṽ‖ → 0, as t → ∞ (see [1]), which proves the firstpart of the Theorem. From [1] we know that ‖Ttṽ‖p ≤ e(γp+1)t‖ṽ‖p. It gives theexponential stability of the semigroup (Tt)t≥0 with D = eCq and ω = − 1p (λ(0)p + 1).AcknowledgementsThe second author acknowledges the support from Białystok Technical University(Grant No. S/WI/1/08).REFERENCES[1] Z. Brzeźniak, A.L. Dawidowicz, On the periodic solution to the von Foerster-Lasotaequation, to appear in Semigroup Forum.[2] A.L. Dawidowicz, On the existence of an invariant measure for a quasi-linear partialdifferential equation, Zeszyty Naukowe UJ, Prace Matematyczne 23 (1982), 117–123.[3] A.L. Dawidowicz, On the existence of an invariant measure for the dynamical systemgenerated by partial differential equation, Ann. Polon. Math. XLI (1983), 129–137.[4] A.L. Dawidowicz, N. Haribash, On the periodic solutions of von Foerster type equation,Universitatis Iagellonicae Acta Mathematica (1999) 37, 321–324.[5] A. Lasota, G. Pianigiani, Invariant measures on topological spaces, Boll. Un. Mat. Ital.(5) 15-B (1977), 592–603.[6] A. Lasota, T. Szarek, Dimension of measures invariant with respect to Ważewska partialdifferential equation, J. Differential Equations 196 (2004) 2, 448–465.[7] K. Łoskot, Turbulent solutions of first order partial differential equation, J. DifferentialEquations 58 (1985) 1, 1–14.[8] R. Rudnicki, Invariant measures for the flow of a first order partial differential equation,Ergodic Theory and Dynamical Systems, 5 (1985) 3, 437–443.Antoni Leon DawidowiczAntoni.Leon.Dawidowicz@im.uj.edu.plJagiellonian UniversityInstitute of Mathematicsul. Prof. Łojasiewicza, 30-348 Kraków, PolandOn periodic and stable solutions of the Lasota equation 461Anna Poskrobkoaposkrobko@wi.pb.edu.plBiałystok Technical UniversityInstitute of Mathematics and Physicsul. Wiejska 45A, 15-351 Białystok, PolandReceived: March 3, 2008.Accepted: October 10, 2008.

Opuscula Mathematica

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On periodic and stable solutions of the Lasota equation in different phase spaces

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Opuscula Mathematica • Vol. 28 • No. 4 • 2008Professor Andrzej Lasota in MemoriamAntoni Leon Dawidowicz, Anna PoskrobkoON PERIODIC AND STABLE SOLUTIONSOF THE LASOTA EQUATIONIN DIFFERENT PHASE SPACESAbstract. We study properties of the Lasota partial diﬀerential equation in two diﬀerentspaces: Vα (Hölder continuous functions) and Lp. The aim of this paper is to generalize theresults of [1].Keywords: partial diﬀerential equations, periodic solutions, stable solutions.Mathematics Subject Classiﬁcation: 35B10, 35B35, 37C75, 47D06.1. INTRODUCTIONWe consider the partial diﬀerential equation∂u∂t+ x∂u∂x= λ(x)u, t ≥ 0, 0 ≤ x ≤ 1 (1.1)with the initial conditionu(0,x) = v(x), 0 ≤ x ≤ 1, (1.2)where v belongs to some normed vector space V of functions deﬁned on [0,1] andλ : [0,1] → R is a given continuous function. Let a semidynamical systemTt : V → Vbe given by the formula(Ttv)(x) = u(t,x),where u is the solution of (1.1), (1.2). It is clear that this unique solution is given bythe formula(Ttv)(x) = u(t,x) = eg(x)e−g(xe−t)v(xe−t), x ∈ [0,1], (1.3)453454 Antoni Leon Dawidowicz, Anna Poskrobkowhereg(x) = −Z 1xλ(s)sdswith the condition Z 10λ(s)sds = ∞. (1.4)We wish to investigate some properties of system (1.3): periodic solutions, strong andexponential stability.Deﬁnition 1.1. A function v0 ∈ V is a periodic point of the semigroup (Tt)t≥0, witha period t0 ≥ 0 iﬀ Tt0v0 = v0. A number t0 > 0, is called a principal period of aperiodic point v0 iﬀ the set of all periods of v0 is equal to Nt0.Deﬁnition 1.2. The semigroup (Tt)t≥0 is strongly stable in V iﬀ for every v ∈ V ,limt→∞Ttv = 0 in V.Deﬁnition 1.3. The semigroup (Tt)t≥0 is exponentially stable iﬀ there exist D < ∞and ω > 0 such thatkTtk ≤ De−ωt, for t ≥ 0.The problem of the chaotic behaviour of a partial diﬀerential equation was consid-ered by Lasota [5], Rudnicki [8], Łoskot [7] and Szarek [6]. In the papers [1–4] therewere described properties of the partial diﬀerential equation, analogical to (1.1), butwith a constant function λ:∂u∂t+ x∂u∂x= γu, t ≥ 0, 0 ≤ x ≤ 1 (1.5)and with the initial conditionu(0,x) = v(x), 0 ≤ x ≤ 1. (1.6)This work has been intended as an attempt at generalizing the results of [1]. In [1]there was described the chaotic and stability behaviour of the suitable semidynamicalsystem(e Ttv)(x) = e u(t,x) = eγtv(xe−t), x ∈ [0,1] (1.7)in diﬀerent phase spaces V . All properties depended on the value γ. We are interestedin ﬁnding a connection between this two equations. It is easy to check that if u ande u are the solutions of equation (1.1) and (1.5), respectively, thene u(t,x) = κ(x)u(t,x), (1.8)whereκ(x) = eR x0λ(0)−λ(s)s ds and γ = λ(0). (1.9)Hence the diagramOn periodic and stable solutions of the Lasota equation 455VTt − − − − → Vmκ y ymκV − − − − →e TtVThis substitution will be a useful tool. It will be used in the proofs of theoremson chaos and stability of system (1.3) in the spaces Vα and Lp.2. PROPERTIES OF THE DYNAMICAL SYSTEM (Tt)t≥0 IN THE SPACE VαLet v be a continous function on [0,1] such that v(0) = 0. For every interval A ⊂ [0,1]and for every α ∈ (0,1], deﬁneHA,α(v) = supx,y∈A,x6=y|v(x) − v(y)||x − y|α .A function v for which HA,α(v) < ∞ is called a Hölder continuous on the interval Awith exponent α. WriteHα = H[0,1],α.Deﬁnition 2.1. Denote by Vα the space of all Hölder continuous functions v on [0,1]with exponent α, vanishing at zero and satisfying the following conditionlimx→0H[0,x],α(v) = 0.Certain properties of system (1.7) in the space Vα have been estabilished. Forγ > α, there exist periodic solutions of problem (1.5) and the set of all periodicpoints is dense in Vα. Strong and exponential stability take place, provided thatγ ≤ α and γ < α, respectively (see [1] for more details).Theorem 2.2. Let|λ(0) − λ(x)| ≤ Cx, C > 0, x ∈ [0,1] (2.1)hold. Then the function u ∈ Vα if and only if e u ∈ Vα.Proof. The assumption u ∈ Vα means that u is a Hölder continuous function withexponent α, vanishing at zero and lima→0 H[0,a],α(u) = 0. This givesH[0,a],α(e u) = supx,y∈[0,a],x6=y|e u(t,x) − e u(t,y)||x − y|α == supx,y∈[0,a],x6=y|κ(x)u(t,x) − κ(y)u(t,y)||x − y|α ≤≤ supx∈[0,a]|κ(x)| · supx,y∈[0,a],x6=y|u(t,x) − u(t,y)||x − y|α ++ supy∈[0,a]|u(t,y)| · supx,y∈[0,a],x6=y|κ(x) − κ(y)||x − y|α .456 Antoni Leon Dawidowicz, Anna PoskrobkoUsing (1.9) and (2.1), we obtain|κ(x)| ≤ eR x0|λ(0)−λ(s)|s ds ≤ eR x0 Cds = eCx.It remains to show that the function κ is Hölder continous. If κ1α apears the Lipschitzfunction, it will mean that κ is Hölder continuous with exponent α. For x ∈ [0,a]|(κ1α(x))0| = e1αR x0λ(0)−λ(s)s ds ·1α·λ(0) − λ(x)x  ≤ e1αR x0 Cds ·Cα  == eCxα ·Cα  ≤ eCaα ·Cα.Summarizing,H[0,a],α(e u) ≤ eCa H[0,a],α(u) + supy∈[0,a]|u(t,y)| ·Cαα!.Therefore, lima→0 H[0,a],α(e u) = 0 and ﬁnally e u ∈ Vα. The rest of the proof runsas before. We can draw the same conclusion for the funcition u, assuming thate u ∈ Vα.Assumption (2.1) will be needed throughout this section and the above theoremwill be crucial for next results.Theorem 2.3. If λ(0) > α ∈ (0,1], then for any t0 there exists such v0 ∈ Vα thatTt0v0 = v0 (2.2)andTtv0 = v0 if and only if t = nt0 for some positive integer n. (2.3)Proof. Let w be an arbitrary Hölder continuous function with the exponent α deﬁnedon the interval [e−t0,1] and satisfying the following conditions:e−g(e−t0)w(e−t0) = w(1), (2.4)∀t ∈ (0,t0) e−g(e−t)w(e−t) 6= w(1). (2.5)Consider the following function v on the interval (0,1] :v(x) = eg(x)e−g(xent0)w(xent0) for x ∈ [e−(n+1)t0,e−nt0].The function v is deﬁned on the whole interval (0,1] =S∞n=0(e−(n+1)t0,e−nt0]and come into being by squeezing the graph of the function w into each interval(e−(n+1)t0,e−nt0].By the assumption of the continuity of w on [e−t0,1], its boundedness follows, i.e.,On periodic and stable solutions of the Lasota equation 457there is M > 0 such that |w(x)| ≤ M for each x ∈ [e−t0,1]. By the above, forx ∈ [e−(n+1)t0,e−nt0], the following estimate holds:|v(x)| = eg(x)e−g(xent0)|w(xent0)| ≤ Meg(x) · supx∈[e−t0,1]e−g(x) ≤ M1eg(x),where M1 = M · supx∈[e−t0,1] e−g(x). From assumption (1.4), limx→0 eg(x) = 0, sowe deduce that v(0) = 0. We obtain the continuous function v deﬁned on the wholeinterval [0,1]. Property (2.2) follows from (2.4), while property (2.3) from (2.5).The assumption λ(0) > α yields e v ∈ Vα, see [1] for details. Under Theorem 2.2,we see at once that v ∈ Vα.Theorem 2.4. For λ(0) > α, the set of periodic points of (1.1) is dense in Vα.Proof. Let v be an arbitrary function belonging to Vα. Deﬁnew(x) = eg(x) e−g(xent0)v(xent0) −∞ Xk=n+1me−g(xekt0)!, (2.6)where m = e−g(e−t0)v(e−t0)−v(1) and x ∈ [e−(n+1)t0,e−nt0]. To show the correctnessof this deﬁnition, it is suﬃcient to make the following observationeg(e−(n+1)t0) e−g(e−(n+1)t0ent0)v(e−(n+1)t0ent0) −∞ Xk=n+1me−g(e−(n+1)t0ekt0)!== eg(e−(n+1)t0) v(1) + m −∞ Xk=n+1me−g(e−(n+1)t0ekt0)!== eg(e−(n+1)t0)e−g(e−(n+1)t0e(n+1)t0)v(e−(n+1)t0e(n+1)t0) −−∞ Xk=n+2me−g(e−(n+1)t0ekt0)!.The function w is continuous and vanishes at 0, which is a consequence of (2.6).Let  > 0. Since v ∈ Vα and w ∈ Vα, there exists such t0 that H[0,e−t0],α(v) < 4 andH[0,e−t0],α(w) < 4. From (1.8) we know that w(x) =e w(x)κ(x), where e w is the periodicsolution of (1.5). For λ(0) > α the set of periodic points of (1.5) is dense in Vα (see[1]), so Hα(v − e w) < 4 and Hα(w − e w) < 4.ThusHα(v − w) ≤ H[e−t0,1],α(v − w) + H[0,e−t0],α(v − w) ≤≤ Hα(v − e w) + Hα(e w − w) + H[0,e−t0],α(v) + H[0,e−t0],α(w) << .This completes the proof.458 Antoni Leon Dawidowicz, Anna PoskrobkoTheorem 2.5. If λ(0) ≤ α and v ∈ Vα, thenlimt→∞Hα(Ttv) = 0.Moreover, if λ(0) < α, then the semigroup (Tt)t≥0 is exponentially stable.Proof. Take any v ∈ Vα. Using Theorem 2.2 and proceeding analogously as in itsproof, we computeHα(Ttv) = supx,y∈[0,1],x6=y|u(t,x) − u(t,y)||x − y|α = supx,y∈[0,1],x6=ye u(t,x)κ(x) −e u(t,y)κ(y)|x − y|α ≤≤ supx∈[0,1] 1κ(x)  · supx,y∈[0,1],x6=y|e u(t,x) − e u(t,y)||x − y|α ++ supy∈[0,1]|e u(t,y)| · supx,y∈[0,1],x6=y 1κ(x) − 1κ(y) |x − y|α ≤≤ eC · Hα(Tte v) + supy∈[0,1]|e u(t,y)| ·Cαα!.We know that Tte v → 0 in Vα for every e v ∈ Vα. The claim limt→∞ Hα(Tte v) = 0 isbased on the results of paper [1]. From the same source, we have derived the estimatekTte vk ≤ e(γ−α)tke vk.|e u(t,y)| = |(Tte v)(y)| = |(Tte v)(y) − (Tte v)(0)| ≤ Hα(Tte v)yα ≤ Hα(Tte v),hence, since λ(0) < α, there follows the exponential stability of the semigroup (Tt)t≥0with D = eC1 + Cααand ω = α − λ(0).3. CHAOS AND STABILITY OF THE SYSTEM (Tt)t≥0 IN THE SPACE LpTheorem 3.1. Assume that∃C,q > 0 ∀x ∈ [0,1] |λ(0) − λ(x)| ≤ Cxq. (3.1)The function u belongs to the space Lp if and only if e u ∈ Lp.Proof. Using (3.1), substitution (1.8) discussed in Section 1 and assuming that u ∈ Lp,we obtainke u(t,x)kp =Z 10|κ(x)u(t,x)|pdx ≤Z 10epR x0|λ(0)−λ(s)|s ds|u(t,x)|pdx ≤≤Z 10eCpq xq|u(t,x)|pdx ≤ eCpqZ 10|u(t,x)|pdx == eCpq ku(t,x)kp < ∞In the same manner, we can establish the inverse implication.On periodic and stable solutions of the Lasota equation 459The above theorem will be signiﬁcant in obtaining the next one. It enables theresults of [1] to be used and generalized. From now on, we assume (3.1).Theorem 3.2. For λ(0) > −1p there exists a periodic solution of (1.1).Proof. For any t0, deﬁne the following function v :v(x) = eg(x)e−g(xent0)w(xent0), x ∈ [e−(n+1)t0,e−nt0], (3.2)v(0) = 0,where w is an arbitrary function from the space Lp. We have shown that such functionv is a periodic solution, so it is suﬃcient to prove that v ∈ Lp. The function v is thesolution of equation (1.1), so we can express it using the function e v, the solution of(1.5), v(x) =e v(x)κ(x). Brzeźniak and one of the authors [1] showed that e v is a periodicsolution and belongs to Lp when γ > −1p. Our assumption and Theorem 3.1 guaranteethe same conclusion for the function v.Theorem 3.3. If λ(0) > −1p, then the set of periodic points is dense in Lp.Proof. Let w ∈ Lp and  > 0. Fix t0 such that"Z e−t00|w(x)|pdx# 1p<2andeCq ke v|| <2, C > 0where e v is a periodic solution of (1.5).The function v is deﬁned by formula (3.2). The function v belongs to the set ofperiodic points due to Theorem 3.2. It is suﬃcient to estimate ||v − w||. Sincev(x) = w(x) for x ∈ [e−t0,1], it is obvious that ||v − w|| = ||(v − w)1[0,e−t0]||, where1[0,e−t0] denotes the indicator of the set [0,e−t0]. Applying the estimate from Theorem3.1 and substitution (1.8), we can assert thatkv − wk ≤ kv1[0,e−t0]k + kw1[0,e−t0]k < .Theorem 3.4. If λ(0) ≤ −1p, and v ∈ Lp thenlimt→∞kTtvk = 0.Moreover, for λ(0) < −1p the semigroup (Tt)t≥0 is exponentially stable on Lp.460 Antoni Leon Dawidowicz, Anna PoskrobkoProof. Let v ∈ Lp be an arbitrary function.kTtvkp =Z 10|u(t,x)|pdx =Z 10 e u(t,x)κ(x) pdx ==Z 10 1κ(x)(Tte v)(x) pdx ≤ eCpq kTte vkp,where C > 0. We know that kTte vk → 0, as t → ∞ (see [1]), which proves the ﬁrstpart of the Theorem. From [1] we know that kTte vkp ≤ e(γp+1)tke vkp. It gives theexponential stability of the semigroup (Tt)t≥0 with D = eCq and ω = −1p(λ(0)p + 1).AcknowledgementsThe second author acknowledges the support from Białystok Technical University(Grant No. S/WI/1/08).REFERENCES[1] Z. Brzeźniak, A.L. Dawidowicz, On the periodic solution to the von Foerster-Lasotaequation, to appear in Semigroup Forum.[2] A.L. Dawidowicz, On the existence of an invariant measure for a quasi-linear partialdiﬀerential equation, Zeszyty Naukowe UJ, Prace Matematyczne 23 (1982), 117–123.[3] A.L. Dawidowicz, On the existence of an invariant measure for the dynamical systemgenerated by partial diﬀerential equation, Ann. Polon. Math. XLI (1983), 129–137.[4] A.L. Dawidowicz, N. Haribash, On the periodic solutions of von Foerster type equation,Universitatis Iagellonicae Acta Mathematica (1999) 37, 321–324.[5] A. Lasota, G. Pianigiani, Invariant measures on topological spaces, Boll. Un. Mat. Ital.(5) 15-B (1977), 592–603.[6] A. Lasota, T. Szarek, Dimension of measures invariant with respect to Ważewska partialdiﬀerential equation, J. Diﬀerential Equations 196 (2004) 2, 448–465.[7] K. Łoskot, Turbulent solutions of ﬁrst order partial diﬀerential equation, J. DiﬀerentialEquations 58 (1985) 1, 1–14.[8] R. Rudnicki, Invariant measures for the ﬂow of a ﬁrst order partial diﬀerential equation,Ergodic Theory and Dynamical Systems, 5 (1985) 3, 437–443.Antoni Leon DawidowiczAntoni.Leon.Dawidowicz@im.uj.edu.plJagiellonian UniversityInstitute of Mathematicsul. Prof. Łojasiewicza, 30-348 Kraków, PolandOn periodic and stable solutions of the Lasota equation 461Anna Poskrobkoaposkrobko@wi.pb.edu.plBiałystok Technical UniversityInstitute of Mathematics and Physicsul. Wiejska 45A, 15-351 Białystok, PolandReceived: March 3, 2008.Accepted: October 10, 2008.

On periodic and stable solutions of the Lasota equation in different phase spaces

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