We study the ”finite extinction phenomenon”(there exists t0 ≥ 0 such that u(t, x) ≡ 0 ∀t ≥ t0, a.e. x ∈ Ω) for solutions of parabolic reaction-diffusion equations of the type ∂u ∂t − k∆u+λb(t)f(u(t − τ, x)) = 0 and ordinary delayed differential equations (k = 0) with a delay term τ > 0

Casal Piga, Alfonso Carlos

Díaz Díaz, Jesús Ildefonso

Vegas Montaner, José Manuel

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–7)An extinction delay mechanism for some evolutionequationsA.C. Casal1, J.I. Dı´az2, J.M. Vegas21 Univ. Polite´cnica de Madrid. E-mail: alfonso.casal@upm.es.2 Dpto. de Matema´tica Aplicada, Facultad de Matema´ticas, Universidad Complutense de Madrid, Plazade Ciencias 3, 28040 Madrid jm vegas@mat.ucm.esPalabras clave: Finite extinction time, delayed feedback controls, delayed ordinary and partialdifferential equationsResumenWe study the ”finite extinction phenomenon”(there exists t0 ≥ 0 such that u(t,x) ≡0 ∀t ≥ t0, a.e. x ∈ Ω) for solutions of parabolic reaction-diffusion equations of thetype ∂u∂t − k∆u+λb(t)f(u(t − τ,x)) = 0 and ordinary delayed differential equations(k = 0) with a delay term τ > 0.1. Introduction.In the last years the “finite extinction phenomenon” (there exists te ≥ 0 such thatu(t,x) ≡ 0 ∀t ≥ te, and a.e. x∈ Ω) has been proved for solutions of suitable parabolicreaction-diffusion equations (usually involving some non-Lipschitz nonlinear terms) on anopen bounded set Ω in RN : see, e.g., [2].The main goal of this work is to show how the finite extinction phenomenon may bethe result of the mere presence of a suitable time-delayed reaction term. We consider somespecial evolution equations on the space X = Lp(Ω)(PA,B)∂u∂t +Au+λb(t)f(u(t− τ,x)) = 0 (0,+∞)× Ω,Bu = 0 (0,+∞)× ∂Ω,u(s,x) = u0(s,x) (−τ, 0)× Ω,where λ > 0, f : R → R is a continuous function, f(0) = 0, b ∈ L1loc(0,+∞), b ≥ 0,A : D(A)→ X is a partial differential operator (in most of the cases assumed to be linear)and Bu = 0 represents the boundary conditions on (0,+∞)× ∂Ω. We want to show that,1A.C. Casal, J.I. Dı´az, J.M. Vegasfor many given f, A and B, it is possible to select some u0, λ and b for which the associatedsolution becomes extinct after a finite time. We anticipate that function b will extinct alsoafter a finite time and so that problem (PA,B) recall some problems formulated in theframework of Control Theory.If b(t) ≡ 0 and A is linear the “finite extinction phenomenon” cannot hold because ofwell-known properties such as the unique continuation property or the strong maximumprinciple. In the case of zero delay τ = 0, extinction in finite time is typical of equa-tions containing a strong absorption term. For instance, in the case of reaction-diffusionequations of the type∂u∂t−∆u+ λ |u|m−1 u = 0 (1)for some λ,m > 0 it is well-known (see e.g. Antontsev, Dı´az and Shmarev[2] and itsreferences) that the finite extinction phenomenon takes place if and only if m ∈ (0, 1) .We point out that a sistematic study about under which non local terms of the generalform G(t, ut) (here ut(s,·) := u(t+s,·) for s ∈ [−τ, 0]) the solutions of ∂u∂t −∆u+G(t, ut)+λ |u|m−1 u = 0 becomes extinct after a finite time was made in Redheffer and Redlinger[7]but always under conditionm ∈ (0, 1). Our point of view is different since we are interestedin the pure memory effects, and no condition of the type m ∈ (0, 1) will be required here.Since in the case of Neumann boundary conditions and spatially constant initial data,u(s,x) = u0(s), we can produce solutions by solving an ordinary delayed differentialequation (ODDE) it is natural to start our study by the consideration of this simpler typeof problems. We start by considering, in Section 2, linear ODDEs of the type A = 0 andf(s) = λs for some λ > 0. We show that if b(t) becomes extinct after a “small” timetb = 2τ , b(t) being inactive (i.e. zero) on [0, τ ], then the solution u(t,x) becomes extinctafter the finite time tb. Nonlinear ODDE requires a separated treatment which is presentedin Section 3. Finally, the application to the study of the finite extinction time for somedelayed partial differential equations is the object of Section 4.It seems interesting to point out that, in contrast with what happens for parabolicsecond order equations without any delay, the comparison principle is not useful in ourcontext. Indeed, it is well known (see, e.g. Pao [6] Chapter 1, Theorem 8.1) that if f isnonincreasing the following general comparison principle holds: given T > 0, if u, u ∈C([−τ, T ] : Lp(Ω)) are sub- and supersolutions of the Dirichlet problem∂u∂t −∆u+λb(t)f(u(t− τ,x)) + g(u(t,x)) ≥ 0, (0, T )× Ω,u(t,x) ≥ 0 ,(0, T )× ∂Ω, u(s,x) ≥ u0(s,x) (−τ, 0)× Ω,(replacing ≥ by ≤ for the case of u) then u ≤ u ≤ u on [−τ, T ] × Ω. But here we areinterested in the opposite case in which f is increasing (otherwise the extinction propertywould require the additional conditions already considered in [7]). It is not difficult tocontruct counterexamples showing that, in that case, the comparison principle fails. Ourmain method of proof will rest on the constants variation formula, and so, independentlyof the comparison principle.2. The linear ODDE case.We begin by considering the case of constant initial data:2An extinction delay mechanism for some evolution equationsLemma 1. Consider the initial value problem{W ′(t) + b(t)f (W (t− τ)) = 0,W (s) =W0(s) for s ∈ (−τ, 0]. (2)where f is an increasing function and b(t) is such thatb(t) ≡ 0 for a.a. t ∈ [tb,+∞) for some tb ∈ (0, τ ]. (3)Assume that W0(s) =Wo for all s ∈ (−τ, 0], withWo = f (Wo)tb∫0b(s)ds. (4)Then, the unique solution W of (2) verifies that W (t) ≡ 0 ∀t ≥ tb.Proof. The existence and uniqueness of a strong solution to (2) can be found in classicalbooks as, for instance, Hale [5]. Integrating on (0, t) for t ∈ (0, τ ] we gett∫0W ′(t)dt =W (t)−Wo = −f (Wo)t∫0b(s)ds.But from (4) we get that W (tb) = 0 and as W ′(t) = 0 for a.e. t ∈ [tb,+∞) we conclude.¥To get some other applications to partial differential delayed equations it would beuseful the following:Lemma 2. Let W (t) be a solution of the delay-differential equation{W ′(t) + λnW (t) + λb(t)W (t− τ) = 0,W (s) = µ(s), for s ∈ (−τ, 0) . .Then W (t) ≡ 0 ∀t ∈ [2τ,+∞) assumed thatb(t) = 0, for t ∈ [0, τ ] ∪ [2τ,∞] , (5)1 = λ2τ∫τb(s)eλnsds. (6)Proof. Let us integrate the equation by the method of steps. In the interval t ∈ [0, τ ],the equation is W ′(t) + λnW (t) = 0 and its solution is W (t) = W (0)e−λnt, and soW (τ) =W (0)e−λnτ =⇒W (0) =W (τ)eλnτ . In the interval t ∈ [τ, 2τ ], we have W (t− τ) =W (0)e−λn(t−τ), for t ∈ [τ, 2τ ] , so the equation isW ′(t)+λnW (t) = −λb(t)W (0)e−λn(t−τ)and the solution isW (t) = e−λnt{c+∫ tτeλns(−λb(s)W (0)e−λn(s−τ))ds}.For t = τ , we obtain the value of W (t) = W (0)e−λnt{1− λeλnτ ∫ tτ b(s)ds} , t ∈ [τ, 2τ ].So, the condition λeλnτ∫ 2ττ b(s)ds = 1 implies W (2τ) = 0. Using that b(t) = 0 for t ∈[2τ,+∞) we conclude that W (t) = 0 for any t ∈ [2τ,+∞).¥3A.C. Casal, J.I. Dı´az, J.M. VegasWe shall show, in Section 4,that the above result is of interest in semilinear pa-rabolic delayed equations. As a matter of fact, the arguments can be applied also tosome homogeneous nonlinear problems such as ∂u∂t − ∆pu + λb(t) |u(t− τ,x)|p−2 u(t −τ,x) = 0, giving rise to the ordinary delayed differential equation W ′(t)+λn |W|p−2W+λb(t) |W (t− τ)|p−2W (t−τ) = 0.This, and other different motivations, justify the relevan-ce of the study of the finite extinction time for nonlinear delayed differential equations.3. The nonlinear ODDE case.Consider the nonlinear delayed differential equations.{u′(t) + F (u(t)) + λb(t)f(u(t− τ)) = 0,u(s) = u0(s) for s ∈ (−τ, 0], (7)with our standing hypothesis b(t) ≡ 0 for a.e. t ∈ [0, τ ] ∪ [2τ,+∞).We will show theexistence of a ”branch.of nonzero solutions which vanish in finite time and bifurcate fromthe zero solution. Since we have no explicit expression for the solutions in this case, wehave to assume some extra regularity on F and f . We use a simple version of a well-knownresult:Theorem 1 (Crandall-Rabinowitz bifurcation theorem [3]). Let H = H(ξ, λ) : R ×R → R be smooth, H(0, λ) = 0 for all λ, ∂ξH(0, λ0) = 0 and ∂λ∂ξH(0, λ0) 6= 0. Thenthere exists a smooth ”bifurcating curve”(ξ∗(ε), λ∗(ε)) for ε near 0 such that ξ∗(0) = 0,λ∗(0) = λ0 and ξ∗(ε) 6= 0 if ε 6= 0. This curve satisfies H(ξ∗(ε), λ∗(ε)) = 0.We shall also need the, so called, Alekesev’s nonlinear variation of constants formula.In 1965 Alekseev [1] proved the following: consider the problemsu′ = −F (u), u(s) = ξ, (8)v′ = −F (u) + g(t, v), v(s) = ξ, (9)where F and g are assumed to be C1. The corresponding solutions will be denoted byu(t, s, ξ) and v(t, s, ξ), respectively. Let U(t, s, ξ) denote ∂ξu(t, s, ξ). Then U(t, s, ξ) sas-tisfies the linear variational equation U ′ = −F ′(u(t, s, ξ))U , U(s) = 1. Then the solutionv(t, ξ) of (9) satisfies Alekseev’s nonlinear integral equationv(t, t0, ξ) = u(t, t0, ξ) +∫ tt0U(t, s, u(s, t0, ξ))g (s, v(s, t0, ξ)) ds (10)In our case, the perturbative case g(t, v) is just −λb(t)f(u(t− τ)).Theorem 2. Assume F (0) = f(0) = 0, F ′(0) = 0, f ′(0) = 1, let β :=∫ 2ττ b(s)ds 6= 0 andlet λ∗ = 1/β. Then, there exists a family of initial data u0(0) = uλ0(0) such thatu′(t) + F (u(t)) + λb(t)f(u(t− τ)) = 0, (11)has a branch of nonzero solutions (λ, u) which bifurcate from (λ∗, 0) and vanish for t ≥ 2τwith∥∥uλ0∥∥ 6= 0, if λ > λ∗.Proof. Since F ′(0) = 0, is is clear that u(t, s, 0) = 0, U(t, s, 0) = 1 for all t, s.. Alekseev’sformula gives for t ≥ τ :v(t, 0, ξ, λ) = u(t, 0, ξ)− λ∫ t0U(t, s, u(s, 0, ξ))b(s)f(v(s− τ, 0, ξ, λ))ds4An extinction delay mechanism for some evolution equationsNow, since b(t) = 0 for t /∈ [τ, 2τ ], v(s− τ, 0, ξ, λ) = u(s− τ, 0, ξ) for τ ≤ s ≤ 2τ . So,v(t, 0, ξ, λ) = u(t, 0, ξ)− λ∫ t0U(t, s, u(s, 0, ξ))b(s)f(u(s− τ, 0, ξ))ds (12)Our goal is to show that the equationv(2τ, 0, ξ, λ) = 0 (13)satisfies the hypotheses the Crandall-Rabinowitz bifurcation theorem. By direct differen-tiation, the relevant derivatives to be computed satisfy the following identities:∂ξv(t, 0, ξ, λ) = U(t, 0, ξ)− λ∫ tτ∂ξU(t, s, u(s, 0, ξ))U(s, 0, ξ)b(s)f(u(s− τ, 0, ξ))ds−− λ∫ tτU(t, s, u(s, 0, ξ))b(s)f ′(u(s− τ, 0, ξ))U(s− τ, 0, ξ)dsWhen ξ = 0, u(s− τ, 0, 0) = 0 and then f(u(s− τ, 0, 0) = 0, so the first integral vanishes.The result is∂ξv(t, 0, 0, λ) = U(t, 0, 0)− λ∫ tτU(t, s, 0)b(s)f ′(0)U(s− τ, 0, 0)ds == 1− λ∫ tτU(t, s, 0)b(s)U(s− τ, 0, 0)dssince f ′(0) = 1. Now, U(t, s, ξ) is a solution of a linear homogeneous equation with variablecoefficients, so it satisfies the property U(t, s, ξ)U(s, z, ξ) = U(t, z, ξ) for z ≤ s ≤ t.Onthe other hand, since u(t, s, ξ) satisfies an autonomous equation, we have u(t, s, ξ) = u(t−s, 0, ξ), and this property is inherited by its derivative U(t, s, ξ) = ∂ξu(t, s, ξ). Therefore,U(t, s, 0)U(s − τ, 0, 0) = U(t, s, 0)U(s, τ, 0) = U(t, τ, 0) = 1. Now we evaluate t = 2τ andobtain∂ξv(2τ, 0, 0, λ) = 1− λ∫ 2ττb(s)ds = 1− λβ (= 0, , if λ = λ∗)Moreover ∂λ∂ξv(2τ, 0, 0, λ) = −β 6= 0. Hence (ξ, λ) 7−→ v(2τ, 0, ξ, λ) satisfies the hypothe-ses of the Crandall-Rabinowitz theorem for λ = λ∗ = 1/β and the proof is finished.Remark 1. Note that conditions (13) can be read in the formu(2τ, 0, ξ) = λ∫ 2ττU(t, s, u(s, 0, ξ))b(s)f(u(s− τ, 0, ξ))ds (14)and represents an implicit condition on ξ(= u0(0)), λ and b(s). This condition makessense once that Alekseev’s nonlinear integral formula holds (for instance, some extensionswithout the condition f ′(0) = 1 can be obtained). We point out that even in the abstractframework there are some suitables semilinear equations for which such a formula remainvalid. We also can show that the formula holds for abstract semilinear equations of theform(AP ){dudt +Au+ F (u) = 0 t ∈ (0, T ),u(0) = u05A.C. Casal, J.I. Dı´az, J.M. Vegaswhere T > 0, A : D(A)→ P(X) is a linear maximal monotone operator on a X = L2(Ω)generating a compact semigroup and F : R→ R is a convex increasing function. In thatcase, the equation satisfied by U is of the type U ′ + AU = −(∂F (u(t, s, ξ)))0U where(∂F (u(t, s, ξ)))0 is the element of the subdifferential ∂F (u(t, s, ξ)) of minimal norm. Thiswould be detailed in a future paper and can be obtained by approximating F by itsYosida-approximation, by applying the results for the associated semilinear equation withLipschitz perturbations and passing to the limit.4. On the finite extinction time for some delayed PDEs.Theorem 3. Consider the problem(PD)∂u∂t −∆u+ λb(t)u(t− τ,x) = 0 (0,+∞)× Ω,u(t,x) = 0 (0,+∞)× ∂Ω,u(s,x) = u0(s,x) (−τ, 0)× Ω,with λ > 0. Let u0 ∈ C([−τ, 0] : L∞(Ω)) be such thatu0(s,x) = µ(s)ϕn(x), a.e. x ∈ Ω (15)with µ ∈ C([−τ, 0]), where ϕn is n-th eigenfunction{ −∆ϕn = λnϕn in Ω,ϕn = 0 on ∂Ω..Assume b such that b(t) ≡ 0 for a.e. t ∈ [0, τ ] ∪ [2τ,+∞) andλeλnτ∫ 2ττb(s)ds = 1. (16)Then, there exists a function W (t) with W (t) ≡ 0 ∀t ∈ [2τ,+∞) such that the solutionu of (PD) is given by u(t,x) =W (t)ϕn(x) a.e. x ∈ Ω and for any s ∈ [0, 2τ ]. Therefore uvanishes identically for t ≥ 2τ.Proof. Consider the function u(t,x) = ϕn (x)W (t). It is a routine matter to check that{ut −∆u+λb(t)u(t− τ,x)) =ϕn (x) (W ′(t) + λnW (t) + λb(t)W (t− τ)).So, by taking W (t) as solution of the ODE with delay{W ′(t) + λnW (t) + λb(t)W (t− τ) = 0,W (s) = µ(s), for s ∈ (−τ, 0) , .we find that u is a solution of (PD) which must coincide with u by uniqueness of solutions.It remains to prove that W (t) ≡ 0 ∀t ∈ [2τ,+∞). But this is precisely the result ofLemma 2. Then, the function u(t,x) vanishes globally in Ω.¥Remark 2. The result also gives the way in which the solution of the Dirichlet problem(PD) reaches the identically zero state. We recall that in the case of semilinear equationswith a strong absorption term, (1) m ∈ (0, 1), it is known that a “dead core” appears6An extinction delay mechanism for some evolution equationsgiving rise to a free (or moving) boundary defined as the boundary of the support of u(t, .)([2]). In fact, under symmetry assumptions on the initial data such a dead core ends beingthe complete domain Ω except a single point ([4]). Our result shows that, for some casesfor which the finite-time extinction arises just by addition of a delay term, the decay tozero is spatially uniform on the whole domain Ω.. This proof also shows the way to agreat variety of possible generalizations to other retarded equations associated to differentlinear problems as, e.g. the ones associated to higher order elliptic operators, the Stokesproblem, etc.Remark 3. If we consider, for instance, the zero controllability problem∂u∂t −∆u = v, (0,+∞)× Ω,u(t,x) = 0, (0,+∞)× ∂Ω,u(0,x) = U0(x), Ω,where we want to find a control v such that u(2τ,x) = 0 for a.e. x∈ Ω, we can use Theorem3 to construct v(t,x) = −b(t)u(t− τ,x). Since v becomes extinct after a finite time tb > 0we recover the typical characteristic of switched controls.Remark 4. Whith slight changes, the above argument can be applied also to some ho-mogeneous nonlinear problems such as∂u∂t−∆pu+ λb(t) |u(t− τ,x)|p−2 u(t− τ,x) = 0.In that case the eigenfunctions are given by{ −∆pϕn = λn |ϕn|p−2 ϕn in Ω,ϕn = 0 on ∂Ω..and the ordinary delayed differential equation becomesW ′(t) + λn |W|p−2W + λb(t) |W (t− τ)|p−2W (t− τ) = 0.Acknowledgments.Research partially supported by the projects MTM2005-03463 of the DGISGPI (Spain)and CCG06-UCM/ESP-1110 of the DGUIC of the CAM and the UCM.Referencias[1] V.M. Alekesev, An estimate for the perturbation of solutions of ordinary differential equations (Rus-sian), Vestnik Moskow, Univ. Ser. I Mat. Meh. 2, 1961, 28-36.[2] S. Antontsev, J. I. Dı´az, S. Shmarev, Energy methods for free boundary problems. Applications tononlinear PDEs and Fluid Mechanics, Birka¨user, Boston, 2002.[3] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8, 1971, 321-340.[4] A. Friedman and M. A. Herrero, Extinction properties of semilinear heat equations with strong ab-sorption . J. Math. Anal. and Appl. 124, 1987, 530 - 546.[5] J.K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977.[6] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.[7] R. Redheffer and R. Redlinger, Quenching in time-delay systems: a summary and a counterexample,SIAM J. Math. Anal., 13, 1984, 1114-1124.7

An extinction delay mechanism for some evolution equations

https://idus.us.es/xmlui/bitstream/handle/11441/34922/An%20extinction%20delay%20mechanism%20for%20some%20evolution%20equations.pdf?sequence=1&isAllowed=y

An extinction delay mechanism for some evolution equations

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla