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ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ФУНКЦИОНАЛЬНЫЙ АНАЛИЗ
INTEGRO-DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2019. Т. 30. С. 99-113
УДК 517.956.4, 517.956.8 MSG 35К08, 45Е10
DOI https://doi.org/10.26516/1997-7670.2019.30.99
On the Behaviour at Infinity of Solutions to Nonlocal Parabolic Type Problems
E. A. Zhizhina1, A. L. Piatnitski1-2
1 Institute for Information Transmission Problems of Russian Academy of Siences (IITP RAS), Moscow, Russian Federation
2 The Arctic University of Norway, Campus Narvik, Norway
Abstract. The paper deals with possible behaviour at infinity of solutions to the Cauchy problem for a parabolic type equation whose elliptic part is the generator of a Markov jump process , i.e. a nonlocal diffusion operator. The analysis of the behaviour of the solutions at infinity is based on the results on the asymptotics of the fundamental solutions of nonlocal parabolic problems. It is shown that such fundamental solutions might have different asymptotics and decay rates in the regions of moderate, large and super-large deviations. The asymptotic formulae for the said fundamental solutions are then used for describing classes of unbounded functions in which the studied Cauchy problem is well-posed. We also consider the question of uniqueness of a solution in these functional classes.
Keywords: nonlocal operators, parabolic equations, fundamental solution, Markov jump process with independent increments.
1. Introduction and Statement of the Problem 1.1. Introduction. The Contact Model
Parabolic type equations with a nonlocal elliptic operator on the right-hand side play an important role in the analysis of a population evolution in models of mathematical biology and population dynamics. The presence of a nonlocal operator on the right-hand side of the equation reflects the fact that the interaction in these models has a nonlocal character. Let us describe one of these models, the so-called continuum contact model in Rd, see e.g. [7-9]. It is a continuous time birth and death Markov process in continuum defined on the space of infinite (but locally finite) configurations 7 € T lying in the space Rd: 7 C Rd. The process is characterized by the birth and death rates. Each point 1: € 7 of a configuration 7 might create an offspring y independently on other points of the configuration. The offspring location is distributed in the space with the density a(x—y) (so-called dispersal kernel), and we assume fRd a(z)dz = 1. In addition any point of the configuration has an independent exponentially distributed random life time determined by the mortality rate m(x) > 0, and in the general case the mortality rate is a spatially inhomogeneous function m(x) > 0. The generator of the dynamics of this process takes the form
lf{ 7) = E / № u y) - Ffr)) dy + E m(*) - •
The case of homogeneous mortality m(x) = k has been studied in details in the paper [7]. The most interesting case is k = 1 - the critical regime, when a family of stationary distributions exist.
One of the remarkable property of the contact model is the fact that the evolution equation on the first correlation function (so-called density of configurations) is decoupled and can be considered separately. That is the case only for the first correlation function, evolutions of the higher order correlation functions have more complicated hierarchical structure involving lower order correlation functions. The evolution problem has the form
^ = Au, u = u(t,x), x € Rd, t > 0, u(0,x) = uq{x) > 0, (1.1) where
Au(x) = —m{x)u{x) + / a{x — y)u(y)dy. (1.2)
J Rd
If m(x) = 1, then the operator A takes the form
Au(x) = —u(x) + / a(x—y)u(y)dy = / a(x—y)(u(y)—u(x))dy. (1.3)
JRd JRd
nonlocal parabolic type problems
101
We notice that correlation functions in the contact model, as well as in other models of the population dynamics, need not vanish at infinity, and in some models they can even grow. Thus to study the behaviour of correlation functions we have to consider the evolution equations (1.1)-(1.2) in suitable classes of bounded or increasing functions.
1.2. Estimates of fundamental solutions to some parabolic
type problems
In this section we consider some important classes of parabolic type equations and give a short review of known results on the asymptotic behaviour of the corresponding fundamental solutions. The fundamental solution of the classical heat equation
dtu — A u = 0,
where A is the Laplace operator in Rd, is given by the Gauss-Weierstrass function
, , 1 ( \x — y |2\ ,
= (L4)
For a general parabolic equation
dtu — Lu = 0,
where L is a uniformly elliptic second-order operator in divergence form, the Aronson estimates, see [1], for the fundamental solution are well-known:
, . C ( \x — y\2\
--—J,
where the sign x means that both < and > inequalities hold with probably different constants c > 0 and C > 0.
The fundamental solution of parabolic type equation
dtu + (~A)a/2u = 0, 0<a<2,
where (—A)a^2 is an integro-differential operator of the form
(—A)"/2 / (x) = Cd,a P-v. f £Mz/Mdy> (1.5)
J rd \x — y\
has been studied using the subordination techniques, see [2; 5]. The following asymptotic relation holds:
1.3. The statement of the problem
In this paper we are concerned with parabolic type equations, where instead of the elliptic differential operator L we consider its nonlocal analog, namely, the convolution type operator A given by
Af(x)= [ (f(y)-f(x))a(x-y)dy, (1.6)
J Rd
where the convolution kernel a(x) is a nonnegative, even, bounded, integrable function with bounded second moments:
a(x) > 0; a(x) = a(-x); a(x) £ L°°(Rd) n L1^), (1.7)
/ a{x)dx = 1, / \x\2a(x)dx < oo. (1.8)
JRd JRd
In particular, condition (1.8) implies that the matrix a = {ov,} with
@ %j — J X{Xj (x ^ x ^ dx
J Rd
is well defined and positive definite. It follows from (1.7) that a(x) £ L2(lRd), and for its Fourier transform a(p) we have:
d(p) £Cb(Rd)riL2(Rd), maxa(j)) = a(0) = 1, d(p) ->■ 0 as \p\ ->■ oo.
Rd
Moreover, we assume that the convolution kernel a(x) has a light tail at infinity:
a(x) < ce~b^P with some b > 0 and p > 1. (1.9)
Since A is a bounded operator in L2 (Rd), its heat semigroup etA admits the following representation:
00 *k 00 *k
etA = e-teta* = e-t y fk a = e-tu e-t у fk a
^ k\ ^ k\
k=0 k=1
This sum contains the singular part e Id and the regular part
OO ^ / \
v(x,t)=e-tJ2tk^1f1. (1.10)
k= 1
Therefore, the fundamental solution of (1.6) has the form
u(x,t) = е~*5(х)+v(x,t), (1-11)
and for any f £ L2 (Rd) the solution to the nonlocal Cauchy problem
dtu-Au = 0, u\t=0 = f (1.12)
has the form
u(x, t) = e-tf (x) + (v * f )(x, t), (1.13)
where v is defined by (1.10). Notice the similarity of the representation (1.13) for the solution of nonlocal problem (1.12) and the Poisson integral for the classical Cauchy problem. In the present work we study unbounded at infinity solutions of problem (1.12) using formula (1.13) and the asymptotical estimates of the function v(x,t). Some particular cases of convolution kernels have been considered earlier in [4]. Our approach applies to the generic convolution kernels that satisfy the above conditions.
It should also be noticed that there is a crucial difference between the nonlocal operators defined in (1.5) and in (1.6). Namely, in contrast with the nonlocal operator in (1.5) the operator A defined in (1.6) has an integrable kernel a(x — y). It is also useful to note the probabilistic interpretation of the function v(x,t). Under conditions (1.7), (1.8) the operator A given by (1.6) is a generator of a continuous time Markov jump process. If this process starts at zero, then formula (1.11) determines transition probabilities of the process at time t, and v(x,t) is the density of the process under the condition that at least one jump has been made.
2. Asymptotic estimates of v(x,t) as t ^ to
The asymptotic behaviour of the function v(x,t) depends crucially on the relation between |x| and t. We consider four regions in space-time (x, t):
1) |x| < rt1/2(1 + o(1)) (standard deviations region)
2) \x\ = rt~ {1 + o(l)), 0 < 5 < 1 (moderate deviations region)
3) |x| = rt(1 + o(1)) (5 = 1) (large deviations region)
4) \x\ = rt~ {1 + o(l)), 5 > 1 ("extra-large" deviations region)
Theorem 1 (see [6]). Assume that a(x) satisfies conditions (1.7) - (1.9). Then for the function v(x,t) the following asymptotic relations hold as t ^ to in regions of standard and moderate deviations:
1) if \x\ < rt? for some r > 0, then
c(a) (&-1x,x)
v(x,t) = y-e--(1 + 0(1)), (2.1)
¿2
where c(a) > 0 is a constant depending on the covariance matrix a;
2) if x = rt~i~( 1 + o(l)) with 0 < 5 < 1 and r € Rd\{0}, then
v(x,t) = e"1^^1)) = (2.2)
3) If x = rt(1 + o(1)) (the region of large deviations), then we get
v{x,t) < e-4r)t{l+o{l)) ^ (2_3)
The rate function Ф(г) possesses the following important properties: Ф(0) = 0, Ф(г) > 0 for г ф О, Ф is a convex function,
= -^"V^X1 + о(1)), as |r| —>• 0. (2.4) In addition, if p = 1, then
Ф(г) = Ь|г|(1 + o(l)), |г| —у oo, (2.5) and if a{x) has a compact support, then
Ф(г) > — |r| In |r| |r| —> oo, (2-6)
¡i
where /л depends on the support of a{x).
If the function a{x) satisfies the following two-sided estimate
C2e~b^P < a{x) < C\e~b^P, p > 1, then the following asymptotic formula holds
v{x,t) = e-4r)t{l+o{l)) ^ (2J)
Here the function Ф(г) forp = 1 is defined by (2.5), and if p > 1, then
Ф(Г) = JL-{b(p - l))1/p|r|(ln M)2^ (1 + o(l)), IrHoo. (2.8) P 1
Theorem 2. In the region of "extra-large" deviations, when \x\ » t, the following estimate holds for all sufficiently large t:
v{x,t) < exp | — с|ж| (In }• (2.9)
Ifa(x) has a compact support, then in the region of "extra-large" deviations and large t:
v(x,t) < exp | — c\x\ In l^lj- (2-10)
It should be noted that the Gaussian form of the asymptotics (2.1) in the region of standard deviations is the immediate consequence of the local limit theorem for processes with independent increments. Formula (2.1) can also be derived from the asymptotic representation of the corresponding Fourier transform, see e.g. [3]. In the moderate deviations region the asymptotics of the fundamental solution still coincide with that in the standard deviations region, but only in the logarithmic order. For the pre-exponential factor we can only state the sub-exponential rate of decay. Crucial modifications
of the Gaussian form of the asymptotics occurs in the region of large deviations, when x = rt, see formulae (2.3), (2.7). It is there, at the distances of order t, that the nonlocal character of the operator A starts to play an important role. As seen from (2.4), the fundamental solution is still close to the Gaussian function for small r, but it differs essentially from the corresponding Gaussian function for sufficiently large r, see (2.5), (2.8), (2.6). In the "extra-large" deviations region this difference is further enhanced. As follows from estimates (2.9), (2.10) the nonlocal fundamental solution v(x,t) has more heavy tail at infinity than the classical heat kernel (1.4).
3. Classes of unbounded solutions
Let us observe that the formula (1.13) makes sense for a wider class of initial functions f(x) than the class L2 (Rd). For the classical heat equation one can take as the initial data a function f(x) growing at infinity. Then using the representation for the solution through the Poisson integral one can conclude, see e.g. [10], that if f(x) is a continuous function satisfying estimate
\f(x)\ <Cebx\ b> 0, (3.1)
then the solution exists as 0 < t < 1/46. Moreover, the solution also satisfies an estimate of type (3.1), and it is unique in this class.
A similar statement holds for the nonlocal parabolic type problems considered in this paper. It is clear that the admissible growth of the initial condition will be determined by the behavior of the fundamental solution at infinity, i.e. in the region of "extra-large" deviations. We need the following lemma.
Lemma 1. There exists a constant c > 0 such that in the region {(x,t) : t > 0, ^ » 1} the following estimate holds
v{x,t) <exp{ -c\x\ (ln)!))^} (3.2)
for p > 1, and
v(x,t) < exp { — c\x\ (in | — |)} (3.3) if a(x) has a compact support.
Proof For the proof of (3.3) we use representation (1.10). According to estimates (3.60)-(3.61) from [6], there exist constants av > 0 and x > 0 such that for all sufficiently large x and all k with 1 < k < ap \x\ we have:
a*k{x) <exp{-J^}.
If к satisfies 1 < к < |ж| (log then
а*к{х) <ехр{ = ехр{ - x|a;|(log Й)^}.
We also have
Ej-k -t -— <1. i k] ~ fc<|x|(log(M))-p
Notice that the relation \x\ » t implies \x\(log p t. If k >
\x\(log (x)) p) then, by the Stirling formula,
<exp{ - k log +k} < exp { - (log (y))" log (y)}
<exp{-^M(log(^-))V}.
Combining the last three estimates yields the desired inequality (3.2).
The proof of (3.3) relies on similar arguments. Since a(-) has a finite support, then
v(x,t) = e 1 V t a, ^ < C\e 4exp { max S0(k,t)\, . . kl L k>n\x\t J
k>fi\x\
where /л is a constant that depends on the size of the support of a(-), and So(k,t) = fcln| + k. One can easily check that So(k,t) is a decreasing function of k, if к > t. Since \x\ » t, we have
\x\
max So(k, t) = So(/j,\x\, t) = —ц\х\ In —(1 + o(l)).
к>ц\х\ t
This yields estimate (3.3). □
Remark 1. It should be noted that in the formulation of Lemma the value of t might be finite and arbitrary small. In addition, for all sufficiently large x the constant с in estimate (3.3) is greater than the corresponding constant с in bounds (2.9) - (2.10).
We now turn to the main result of the paper.
Theorem 3. Let conditions (1.7) - (1.9) on the function a(x) be satisfied, and assume that the initial condition f{x) is a continuous function such that i
\f(x)\ < ХеФНМ*!)255", with 0 < 26 < с (3.4)
ifp > 1, or
\f(x)\ < #eftNlnN, with 0 < 26 < c, (3.5)
in the case, when a(x) has a compact support. Here c is the same constant as in (2.9) or (2.10), respectively.
Then for any t > 0 there exists the solution of Cauchy problem (1.12) defined by formula (1.13). This solution satisfies the upper bound
\u{x,t)\ < Kp(i)ec>l(ln|a;|)V, (3.6)
as p > 1, or
\u(x,t)\ < KooCOe^l111^! (3.7)
in the case, when a{x) has a compact support, where c is a constant such that 2b < c < c.
Moreover, the solution of the Cauchy problem is unique in the class of functions that satisfy growth condition (3.6) or (3.7), respectively.
Proof. First we prove the existence of solution of the Cauchy problem in the class of functions that satisfy estimate (3.6) (or (3.7)). As follows from representation (1.13) it is sufficient to estimate (v* f)(x,t). In what follows we consider the case of a{x) with a compact support. If the function a{x) meets the general bound (1.9) with some p > 1, the reasoning will be
c
similar. We first estimate (v * f){x,t) for \x\ > xtc-2b with k = x{c,b) =
c+2b 2^2b;
(y*f)(x,t)= v{z,t)f{x - z)dz (3.8)
J Rd
= J v(z,t)f(x — z)dz + J v(z,t)f(x — z)dz. M<M M>M
For the first integral in (3.8) we have
J v{z,t)f{x- z)dz< J v{z,t)Keb\x-z\Xn\x~z\dz M<M M<M
<* / ,(*,<)«*■"■■*<*< / M,
M<M M<M
where 2b < c < c. We have used here the inequality \x — z\ < 2\x\ that is valid for \z\ < \x\, and the estimate |a;|ln2|a;| < (1 + e)\x\ In\x\, that holds for sufficiently large \x\ and small e > 0.
For estimating the second integral on the right-hand side in (3.8) we use the inequality \x — z\ < 2\z\ and the asymptotic formula (2.10) for v(z,t)
С
in the region of "extra-large" deviations \z\ > \x\ > xtc-2b:
J V(z,t)f(x-z)dz< J p-c\z\\n\l\Ke2b\z\\n2\z\dz_
N>M
с+2Ъ с
If Ы > 2С-2Ь tc~2b, then
-c\z\\n2\z\+2b\z\\n2\z\ + c\z\\n2t = -\z\((c-2b) In\2z\ -cln2i) < -a\z\
(3.9)
with some a > 0. Consequently,
J v(z,t)f(x — z)dz < K J e~alzldz = K1(t) <oo. (3.10)
\z\>xt
C
Thus, for \x\ > xtc-2b the following estimate holds:
\u{x,t)\ < K!(i)ec>|ln|:c| (3.11)
with 2b <c< c.
c
In the case, when \x\ < xtc-2b, we estimate each of integral on the right-hand side of (3.8) separately. For the first integral we get the same estimate as above:
J v{z,t)j{x-z)dz < J v(z,t)Keblx-z^x-zldz<Ked^ln^.
\z\<\x\ \z\<\x\
We divide the second integral into two integrals:
J v(z,t)f(x - z)dz (3.12)
M>M
= / v(z,t)f(x — z)dz + / v(z,t)f(x — z)dz.
\x\<\z\<xtr~-2b \z\>xtc
-2b
Considering the inequality \x — z\ < 2\z\, for the first integral on the right-hand side we obtain
j v(z,t)e2b\z\ln2^dz < jv{z^t)dz =
\x\<\z\<xt^ Rd
The second integral admits the same bound as above:
J v(z,t)f(x - z)dz < Ki(t).
Thus, for \x\ < 25 we aiso obtain the desired estimate (3.11).
In the case of kernels a(-) satisfying condition (1.9) the existence of a solution in the class of functions for which estimate (3.6) holds can be proved in a similar way. There is only one difference. Namely, in this case we should show that for all sufficiently large \z\ the following inequality holds:
-c\z\(\n^-)q+ 2b\z\(\n2\z\)q < -a\z\, q = 0,1). (3.13)
This inequality ensures that the integral in (3.10) is finite. To justify (3.13) it suffices to show that for|z| > nt1 with some x > 0 and 7 > 0 we have
(inM^l^l)«. (3.14)
The validity of the latter inequality can be easily checked if we let 7 = (l-(f )1/?)"1 and k = 2^.
The next step is to prove the uniqueness of solution of the Cauchy problem for nonlocal parabolic equation. The proof is based on Holmgren's principle and follows the line of Section 7.7 in [10]. Let to > 0, and assume that u(x, 0) = 0, x € Rd. We want to prove that this initial condition implies that u(x,t) = 0 in the whole strip [0,to) x Consider the adjoint Cauchy problem
dw f 1
-q^ = ~ J d a(x ~ v)(w(v) ~ w{x))dy, w\t=to =
with the terminal condition tp(x), where tp € V(Rd) = Cq°(\Rd) belongs to the space of smooth functions with compact support. The solution of this problem admits the representation similar to that in (1.13). It reads
w(x,t) = e'^-^ipix) + (v *ip)(x,t0-t), t<to, (3.15)
where the function v(x,t) was defined by (1.10). Since tp has a compact support, the solution (3.15) satisfies
\w(x,t)\<C^ max v(x — y,to—t) (3.16)
y€supp if)
for t < to and \x\ > R > 0, where R is sufficiently large. From (3.3), (3.16) and Remark 1 it follows that
K,,t)|<C,exp{-CN(lnmax{Jjo_n)}, 0 < t < to- (3.17)
The corresponding estimate also holds in the case of a(x) with a light tail at infinity with p > 1.
Thus, the estimate (3.17) implies that the integral
(■u(x,t), w(x,t)) = / u(x,t)w(x,t)dx
J Rd
exists and converges uniformly for 0 < t < to- Moreover, the integrals obtained by replacing u and w with their derivatives with respect to t also converge uniformly for 0 < t < to. Therefore, the function
X(t) = (u(x,t), w(x,t))
is continuous at t € [0, io)- Our assumption u(x, 0) = 0, x € Rd, implies that %(0) = 0. The function %(t) is differentiable on the intervbal t € (0, to), and by the symmetry condition on a(x — y) we have
dt JRd v at ot /
= / / a(x — y)(u(y,t) — u(x,t))w(x,t)dydx (3.18)
JRd JRd
— / a(x — y)(w(y,t) — w(x,t))u(x,t)dydx = 0.
JRd JRd
Thus, %(i) = const = 0.
The formula (3.15) for w(x,t) yields
/ u(x,to)ip(x)dx = lim / u(x,t)w(x,t)dx. JRd t-rto—o JRd
Since
lim / u(x,t)w(x,t)dx = lim %(i) = 0,
i-s-io-0 JRd t-^to-0
then fRd u(x, to)ip{x)dx = 0 for every ф e V(Rd). Consequently, u(x, t) = 0 for 0 < t < to- □
4. Conclusions
In this work we described some classes of initial conditions for which the nonlocal parabolic problems studied here are well-posed. It was shown that the initial conditions of exponential and even slightly stronger growth at infinity are admissible. Moreover, the critical growth condition is characterized by the behaviour at infinity of the convolution kernel of the corresponding nonlocal operator. In particular, it follows from our estimates that the class of admissible initial conditions for the studied here nonlocal Cauchy problem is more narrow than that for the classical heat
equation. Such a difference between the structures of the classes of admissible initial conditions is caused by the fact that the fundamental solution of the nonlocal problem decays at infinity slower than the usual heat kernel, the difference in the behaviour becoming apparent at the distance of order t.
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Elena Zhizhina, Doctor of Sciences (Physics and Mathematics), Leading Scientific Researcher, Institute for Information Transmission Problems RAS, build. 1, 19, Bolshoy Karetny per., Moscow, 127051, Russian Federation, tel.: 7 495 650 4225, e-mail: [email protected]
Andrey Piatnitski, Doctor of Sciences (Physics and Mathematics), Professor, Institute for Information Transmission Problems of RAS, build. 1, 19, Bolshoy Karetny per., Moscow, 127051, Russian Federation; Arctic University of Norway, Campus Narvik, P.O.Box 385, 8505 Narvik, Norway, tel.: 7 495 650 4225, e-mail: [email protected]
Received 30.10.19
0 поведении на бесконечности решений нелокальных задач параболического типа
Е. А. Жижина1, А. Л. Пятницкий1'2
1 Институт проблем передачи информации РАН, Москва, Российская Федерация
2 Арктический университет Норвегии, кампус Нарвик, Норвегия
Аннотация. Изучается возможное поведение на бесконечности решений задачи Коши для уравнений параболического типа, в которых в качестве эллиптического оператора берётся генератор марковского скачкообразного процесса, т. е. оператор нелокальной диффузии. Исследование поведения решений на бесконечности базируется на асимптотике фундаментального решения нелокальных параболических задач. Показано, что такое фундаментальное решение имеет разную асимптотику и скорость убывания в областях умеренных, больших и супер-больших уклонений. На основании этих асимптотических формул описаны классы неограниченных функций, в которых корректны рассматриваемые задачи Коши. Обсуждается также единственность решения в этих классах функций.
Ключевые слова: нелокальные операторы, параболические уравнения, фундаментальное решение, марковский скачкообразный процесс с независимыми приращениями.
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Елена Анатольевна Жижина, доктор физико-математических наук, ведущий научный сотрудник, Институт проблем передачи информации им. А. А. Харкевича РАН, Российская Федерация, 127051, г. Москва, Большой Каретный переулок, 19, стр. 1. тел.: +7 (495) 6504225, e-mail: [email protected]
Андрей Львович Пятницкий, доктор физико-математических наук, профессор, Институт проблем передачи информации им. А. А. Харкевича РАН, Российская Федерация, 127051, г. Москва, Большой Каретный переулок, 19 стр. 1; Арктический университет Норвегии, Норвегия, кампус Нарвик, 8505, Нарвик, тел.: +7 (495) 6504225, e-mail: [email protected]
Поступила в редакцию 30.10.19
