# On Solutions to a Class of Functional Differential Equations with Time-Dependent Coefficients

Authors: Muhammad Mohsin; Syed Touqeer H. Shah; Ali A. Zaidi
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we study initial boundary value problems that involve functional (nonlocal) partial differential equations with variable coefficients. These problems arise in cell growth models with symmetric and asymmetric modes of division. We determine the general solution to the symmetric cell division problem for a certain class of coefficients and establish the convergence of solutions to a large time asymptotic solution. The existence of a steady size distribution (SSD) solution for an asymmetric cell division problem is established and is shown to be the large time-attracting solution for a certain class of coefficients. The rate of convergence of solutions towards the SSD solution is affected by the choice of coefficients and remains unaffected by the asymmetry in cell division. The uniqueness of solutions to the initial boundary value problem is also established.


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## Body

## 1. Introduction

Functional partial differential equations, subject to appropriate initial and boundary conditions, arise in various population models. A population model for animals reproduced by fission was given by Sinko and Streifer [1, 2]. Their model was used in the context of cell growth and division, among others, by Hall and Wake [3]. A size structured model, where the size is the DNA content or mass, entails a pantograph-type first-order partial differential equation (PDE)
(1)∂∂tϑx,t+∂∂xγx,tϑx,t=α2σαx,tϑαx,t−σx,tϑx,t,subject to an initial distribution
(2)ϑx,0=ϑ0xand the boundary condition
(3)limx⟶0+γx,tϑx,t=0for t>0. Here, ϑx,t is the number density of the cells of size x at time t, σx,t is the frequency of division of the cells, and γx,t is the growth rate of the cells. The functional term in the PDE (1) is a consequence of symmetric cell division in which α>1 daughter cells of equal size x are produced when a the cell of size αx divides. The asymmetric cell division, in which a cell divides into two different sizes of daughter cells, say sizes α and β, yields an initial value problem that involves a PDE, with two functional terms, of the form
(4)∂∂tϑx,t+∂∂xγx,tϑx,t=ασαx,tϑαx,t+βσβx,tϑβx,t−σx,tϑx,t,along with conditions (2) and (3). The assumption that division does not cause any loss of DNA content or mass leads to
(5)1α+1β=1,so that α>2>β>1. These cell growth and division models are thus initial boundary value problems that involve a functional PDE.The nonlocal terms imply a dearth of analytical techniques for solving such problems, even for a restricted choice ofσ and γ. The cell growth model (1) and (2) has been studied for specific choices of division and growth rates. A separable solution to the problem was found for constant σ and γ [3]. Perthame and Ryzhik [4] showed that this separable solution is the long time asymptotic solution towards which solutions to the PDE converge exponentially in time. The separable solutions were called “steady size distributions (SSDs)” by Hall and Wake. The full problem was then solved by Zaidi et al. [5]. Hall and Wake [6] also studied the case of linear growth rates γx,t=gx and σx,t=bxr, where r>0, b>0, and g>0, and found separable solutions to the initial boundary value problem. The separable solution in this case, however, is not the long time asymptotic solution since the long time asymptotic solution involves time-dependent oscillations [7]. The full problem was solved recently [8]. The division function σx,t=bHx−c, where b and c are positive numbers and H is the Heaviside function, was considered by van-Brunt et al. [9] for constant growth rate. They showed the existence of a steady size distribution solution. For a certain class of functions σx, the existence of a long time asymptotic solution for γx,t=g, where g>0, was established in the context of growth-fragmentation equation [4, 10–12].For asymmetric binary cell division, the PDE (4), together with conditions (2) and (3), was studied for coefficient coefficients ([13, 14]) and for γx,t=gx and σx,t=bxr, where r>0, b>0, and g>0 [15], and separable solutions were determined. The full problem, even for constant coefficients, has not been solved hitherto. Recently, the existence of a unique solution to the problem for constant coefficients has been established [16].The major focus, hitherto, has been on the cell growth PDEs whereσ and γ are either constant or functions of x alone. This makes the separation of variables possible, and the separable solution in most cases corresponds to the long time asymptotic solution. However, time-dependent coefficients classically appear in biology. Michel et al. [11] studied time-varying coefficients in the context of renewal equation with T-periodic death and birth rates. In this paper, we investigate time-dependent growth and division rates. In the next section, we obtain the general solution to PDE (1), subject to conditions (2) and (3), for the case where the growth and division rates have same time dependence, i.e., σx,t=brt and γx,t=grt, where b, g, and r>0. We also find the general solution to PDE (1) for different time dependence of growth and division rates. Specifically, we consider σx,t=btv and γx,t=gtu, where u≥0 and v≥0. We then study the large time asymptotics of solutions for v=0 and u≥0.In Section3, we determine a probability density function eigenvalue and a separable solution to the PDE (4), subject to conditions (2) and (3) for a certain class of division and growth rates. We show that this separable solution is the long time asymptotic solution. We also show that if the problem has a solution, then it is unique.


## 2. General Solution for Symmetric Division

In this section, we find solutions to PDE (1), subject to conditions (2) and (3), for γx,t=grt and σx,t=brt, where g>0, b>0, and r>0 for all t>0. This converts PDE (1) to
(6)∂∂tϑx,t+grt∂∂xϑx,t+brtϑx,t=bα2rtϑαx,t,where α>1. Let
(7)Rt=∫rtdt.Using transformations(8)ϑx,t=exp−bRtϑ~x,t,x=gx^,equation (6) reduces to
(9)∂∂tϑx,t+rt∂∂xϑx,t=bα2rtϑαx,t.The above equation was solved forrt≡1 by Zaidi et al. [5]. Motivated by their analysis, we first find solutions in the region x≥Rt and then extend the solution to 0<x<Rt. Let
(10)N0x,t=ϑ0x−Rt,and fork≥1,
(11)∂∂tNk+rt∂∂xNk=bα2rtNk−1αx,t,where Nk satisfy
(12)Nkx,0=0.This can be solved using method of characteristics with characteristic coordinatesξ=t and η=x−Rt. For k≥1, we have
(13)Nk=∑j=0kdk,jTkwk,jx,t.Here,(14)d0,0=1,dk,j=bα2dk−1,j−1αk−jαj−1,dk,0=−∑j=1kdk,j,wk,jx,t=αk−jαjx−Rt,for k=1,2,⋯, and j=1,⋯k. Moreover, Tk is an antiderivative of Tk−1. It can be verified that the series defining Qx,t=∑k=0∞Nkx,t is the solution to (9) and converges uniformly in any set of the form x,t∈W0:t≤B, where B is any fixed number lying in the region 0<x<Rt. Let mx,t=∫x∞ϑξ,tdξ. Then, PDE (9) reduces to
(15)∂∂tmx,t+rt∂∂xmx,t=bαrtmαx,t.Using a similar approach, the solution to (15) can be gleaned in the region x≥Rt.We extend the solutions of (9) in the region 0<x<Rt by constructing a sequence of wedges in such a ways that if a point x,t lies in a given region, the αx,t is in a region we already know the solutions (see Figures 1 and 2). This makes the functional PDE local and thus able to be solved by using standard methods.Figure 1
The construction of wedges for the casert=t and α=2.Figure 2
The construction of wedges for the casert=1/1+t and α=2.Let(16)Wn=x,t:Rtαn≤x≤Rtαn−1,and let solution to differential equation in the wedge Wn be denoted by hn for n≥1. Also, let
(17)Pn=∑k=n∞∑j=nkck,jMkwk,j,for n≥0, where
(18)c0,0=1,ck,j=bαck−1,j−1αk−jαj−1,ck,0=−∑j=1kck,j,for k=1,2,⋯, and j=1,⋯,k. We can mimic the analysis in [5]. This yields the solution
(19)mx,t=h0=P0ifx,t∈W0,h1=P1+G0ifx,t∈W1,⋮hn=Pn+G0w0,0+∑k=1n−1bαk∏m=1kαm−1Gkwk,kifx,t∈Wn,⋮whereG0 is an arbitrary function that can be determined from the boundary condition (3).The analysis of Zaidi et al. [5] can also be mimicked to determine the large time asymptotic of (19). It can be shown that solution to (9) converges to separable solution as t⟶∞. That is,
(20)nx,t=e−bαRt∑k=0∞ake−bαkx,where ak are determined as in [5].Even if the growth and division rates have different time dependence, (19) still remains the solution to the PDE (9). For instance, if γx,t=gtu and σx,t=btv, where b,g,u, and v are positive constants, (19) solves the PDE
(21)∂∂tmx,t+tu∂∂xmx,t=bαtvmαx,t,along with
(22)m0,t=expbαtvv+1,(23)mx,0=m0x=∫x∞ϑ0σdσ.The large time dynamics, however, differ markedly from the same time dependence case. This is because separable solutions do not exist if the growth and division rates have a different time dependence.We study the long time dynamics whenu>0 and v=0.The convergence ofPn implies Pn⟶0 as n⟶∞. Consequently, the solutions (19) and (22) yield
(24)ebαt=h0,t=G0−tu+1u+1+∑k=1∞bαk∏m=1kαm−1Gk−tu+1u+1,so that
(25)ebα−σu+11/u+1=G0σ+∑k=1∞bαk∏m=1kαm−1Gkσ,where σ=−tu+1/u+1. Since ebα−σu+11/u+1=∑k=0∞−1kbαk/k!u+1k/u+1σk/u+1, the Laplace transform yields
(26)hs≔∑k=0∞−1kbαku+1k/u+1k!sk+u+1/u+1Γk+u+1u+1,fors>0.Consequently, the Laplace transform of (24) gives hs=gs1+∑k=1∞bα/sk/∏m=1k1−qm, so that the Euler’s identity [5] yields
(27)hs=gs∏k=0∞1+bαks,where gs is Laplace transform of G0σ.Thus,(28)gs=hs·11+b/s1+b/αs1+b/α2s⋯.Let(29)Js≔11+b/s1+b/αs1+b/α2s⋯.Then,Js has simple poles at s=−b/αn for n=0,1,2,⋯, and consequently, Mittag-Leffer theorem implies
(30)Js=a0s+b+a1s+b/α+⋯+Zs,whereZs is the entire function. The inverse transform of Js is
(31)J^σ=∑n=0∞ane−bα−nσ,where an=Ress=−b/αnJs.Now,(32)a0=Ress=−bJs,=lims⟶−bs+bJs,=−b∏k=1∞1−1/αk=R~b,α.Consequently, fork≥1,
(33)ak=−1k∏m=1kR~b,α.Hence,(34)J^σ=R~b,αe−bσ+∑k=1∞−1k+1∏m=1kαm−1e−b/αkσ.Equations (27) and (30) imply that
(35)G0σ=ebα−σu+11/u+1∗J^σ,where ∗ represents convolution. This gives
(36)G0σ=R~b,αM1σ+N1σ,where
(37)M1σ≔∫0σebα−θu+11/u+1·e−bσ−θdθ,N1σ=∑k=1∞−1k+1∏m=1kαm−1∫0σe−b/αkσ−θ·ebα−θu+11/u+1dθ,so that, forn≥1,
(38)Gnσ=R~b,αMnσ+Nnσ,where Mnσ is an anti-derivative of Mn−1σ and Nnσ is an anti-derivative of Nn−1σ.


## 3. Steady Size Distribution Solutions and Large Time Asymptotics for Asymmetric Division

Separable solutions are of central interest since these usually correspond to the steady size distribution solutions which attract solutions to the PDE for large time. We show that ifσ and γ are separable with same time dependence, i.e., σx,t=brtmx and γx,t=grthx, where b and g are positive numbers and r, h, and k are positive functions for all t>0 and x>0, then variables can be separated in principle. Consequently, we consider the PDE
(39)∂∂tϑx,t+grt∂∂xhxϑx,t=αbrtmαxϑαx,t+βbrtmβxϑβx,t−brtmxϑx,tand solutions of the form
(40)ϑx,t=ωtρx,where ϱ is required to be a probability density function with ∫0∞ϱxdx=1 and ωt represents the total population of cells of all sizes at a given time t. The separable form (40) and PDE (39) give
(41)ω′tϱx+grth′xωtϱ′x+grthxωtϱ′x+brtmxωtϱx=αbrtmαxωtϱαx+βbrtmβxωtϱβx,which, dividing by rtωtϱx, yields
(42)ωt=ce−ΛRt,for some constant c>0, separation constant Λ (to be determined), and
(43)Rt=∫rtdt.The functionϱx satisfies
(44)ghxϱ′x+gh′x+bmx−Λϱx=αbmαxϱαx+βbmβxϱβx,along with the conditions
(45)limx⟶0+ghxϱx=0,limx⟶∞ghxϱx=0.Equation (44) is a pantograph-type equation with two nonlocal terms. It appears in various applications including the absorption of light in the Milky Way [17] and internet protocols [18]. Although there are no general methods of solving (44), solutions have been obtained for constant coefficients [14] and for hx=x and mx=xr, where r>0 [15]. In both instances, the solution entails a positive Dirichlet series of the form
(46)∑ù=0∞∑ν=0∞dù,νexp−αsùβsνrfx,where ck are coefficients, s and r are constants, and f satisfies
(47)fαx=αsfx.Zaidi and van-Brunt [15] showed that for any positive solution to (27), Λ<0.We show that ifh is not constant or h is not a linear monomial, then (44) does not possess a Dirichlet series solution of the form (46). To establish this, we suppose on the contrary that a Dirichlet series solution of the form (46) to the PDE (44) exists. Equations (44) and (46) give
(48)∑ù=0∞∑ν=0∞dù,νexp−αsùβsνrfxgh′x+∑ù=0∞∑ν=0∞−αsùβsνrdù,νexp−αsùβsνrfxghxf′x+∑ù=0∞∑ν=0∞dù,νexp−αsùβsνrfxmx−Λ∑ù=0∞∑ν=0∞dù,νexp−αsùβsνrfx=α∑ù=0∞∑ν=0∞dù,νexp−αsù+1βsνrfxmαx+β∑ù=0∞∑ν=0∞dù,νexp−αsùβsν+1rfxmβx.Balancing the coefficients yields(49)gh′x−rhxf′x+mx−Λ=0,so that
(50)fx=1rg∫h′xhxdx+∫mxhxdx−Λ∫1hxdx.For constanth and m [13], fx=a1x, for some constant a1, so that f satisfies (47). Also, for hx=x and mx=xs, s>0 [15]; the eigenvalue Λ=−g, so that fx=a2xs/s for some constant a2. Consequently, f satisfies (47). It may be possible that mx~−h′x, but since Λ<0, balancing coefficients in (48) requires mx and h′x to be constants. Consequently, if h is not constant or a linear monomial, the first of these integral yields a logarithmic function which is not a homogeneous function in general. Hence, fαx≠αsfx for any s∈ℤ, and equation (44) does not possess a Dirichlet series solution of the form (46).Forh≡1 and m≡1, the eigenvalue Λ and the corresponding solution to (44) have been determined [13]. The eigenvalue is
(51)Λ=2b,and solution ρ to the corresponding eigenvalue is
(52)ρx=∑ù=0∞∑ν=0∞dù,νe−2b/gαùβνx,where dù,ν’s are determined as in [13].We now show that the separable solution to ((39)) (((2)) and ((3)) obtained in Section 3 forσx,t=brtandγx,t=grt) is the large time-attracting solution for a certain class ofrt. For this choice of coefficients, PDE (39) reduces to
(53)∂∂tϑx,t+grt∂∂xϑx,t=αbrtϑαx,t+βbrtϑβx,t−brtϑx,t.For constant coefficients and symmetric cell division, Zaidi et al. [5] derived the large time-attracting solution from solutions to the PDE (1). The solution turned out to be the separable solution.Since the full problem ((53)) (((2)) and ((3))) has not been solved for any choice of coefficients, we employ the tools developed by Perthame and Ryzhik [4] to study the long time dynamics. The analysis in [4], however, is valid for one nonlocal term and for constant coefficients, and certain arguments break down because of the t dependence of coefficients. We can, nonetheless, extend their analysis to a certain class of time-dependent coefficients and asymmetric cell division.Theorem 1.
Letϑ be a solution to (53) that satisfies (2) and (3) and Rt and ρ be defined by (43) and (52), respectively. If rt is monotonically decreasing and Rt is bounded at t=0 and Rt goes to infinity as t tends to infinity, then
(54)limt⟶∞∫0∞ϑx,te−bRt−kρx=0,where k=e−bR0∫0∞ϑ0xdx.Proof.
Let(55)ux,t=ϑx,te−bRt−kρx.
Then,u satisfies the PDE
(56)∂∂tux,t+grt∂∂xux,t+2brtux,t=αbrtuαx,t+βbrtuβx,tand the conditions
(57)u0,t=0,limx⟶∞ux,t=0.Also, the integration of (56) with respect to x from 0 to ∞, along with condition (2), yields
(58)∫0∞ux,tdx=0.The transformation(59)ζ~x,t=∫0xus,tdsyields
(60)∂∂tζ~x,t+grt∂∂xζ~x,t+2brtζ~x,t=brtζ~αx,t+brtζ~βx,t,along with the conditions
(61)ζ~0,t=0,(62)limx⟶∞ζ~x,t=0.Equation (60) gives
(63)∂∂tζ~x,tebRt+grt∂∂xζ~x,tebRt+brtζ~x,tebRt=brtζ~αx,tebRt+brtζ~βx,tebRt,which, multiplication by sgnζ~, yields
(64)∂∂tζ~x,tebRt+grt∂∂xζ~x,tebRt+brtζ~x,tebRt≤brtζ~αx,tebRt+brtζ~βx,tebRt.Integrating (64) from 0 to ∞ with respect to x and using conditions (61), (62), and (5) yields
(65)ddt∫0∞ζ~ebRtdx≤0,which shows that ∫0∞ζ~ebRtdx is a decreasing function in time, so that
(66)∫0∞ζ~ebRtdx≤ebR0∫0∞ζ~0xdx,where ζ~0x=ζ~x,0 and R0=R0.
(67)P¯x,t=ζ~x,tebRt.Then, (63) becomes
(68)∂∂tP¯x,t+grt∂∂xP¯x,t+brtP¯x,t=brtP¯αx,t+brtPβx,t,which, differentiating with respect to t, gives
(69)∂∂tκx,t+gr′t∂∂xP¯x,t+grt∂∂t∂P¯∂x+br′tP¯x,t+brtκx,t=br′tP¯αx,t+br′tP¯βx,t+brtκαx,t+brtκβx,t,where
(70)κ=∂P¯∂t.Equations (69) and (68) give
(71)∂∂tκx,t+grt∂∂xκx,t−r′trtκx,t+brtκx,t=brtκαx,t+brtκβx,t.Equation (71) can be multiplied with sgnκ. This gives
(72)∂∂tκx,t+grt∂∂xκx,t−r′trtκx,t+brtκx,t≤brtκαx,t+brtκβx,t,which, integrating with respect to x from 0 to ∞ and using (5) and the fact that κ0,t=0=κ∞,t, yields
(73)ddt∫0∞κdx≤0,which shows that ∫0∞κdx≤0 is decreasing in t, so that
(74)∫0∞κdx≤∫0∞κ0xdx,where
(75)κ0x=κx,0=∂P¯∂tt=0.SinceP¯=ζ~ebRt, we have
(76)κ0x=∂∂tζ~ebRtt=0=ζ~0xbr0ebR0+ebR0∂ζ~∂tt=0,where r0=r0 and R0=R0. Equations (60) and (59) imply
(77)∂ζ~∂tt=0=r0−gu0x−2bζ~0x+bζ0αx+bζ~0βx,so that
(78)κ0x=ebR0r0−gu0x−2bζ~0x+bζ~0αx+bζ~0βx+bζ~0x.Equations (74), (78), and (5) give
(79)∫0∞κdx=∫0∞∂∂tζ~ebRtdx≤ebR0r0∫0∞gu0x+2bζ~0x+bζ~0αx+bζ~0βx+bζ~0xdx=ebR0r0∫0∞gu0x+4b∫0∞ζ~0xdx.Equations (59) and (63) give
(80)ux,t=∂∂xζ~x,t=e−bRtgrt−∂∂tζ~x,tebRt−brtζ~x,tebRt+brtζ~αx,tebRt+brtζ~βx,tebRt,so that
(81)∫0∞ux,tdx≤e−bRtgrt∫0∞∂∂tζ~x,tebRtdx+brt∫0∞ζ~x,tebRtdx+brt∫0∞ζ~αx,tebRtdx+brt∫0∞ζ~βx,tebRtdx,which, using (5), gives
(82)∫0∞ux,tdx≤e−bRtgrt∫0∞∂∂tζ~x,tebRtdx+2brt∫0∞ζ~x,tebRtdx.Inequalities (82), (79), and (66) give
(83)∫0∞ux,tdx≤e−bRtgrtebR0r0∫0∞gu0xdx+4b∫0∞ζ~0xdx+2brtebR0∫0∞ζ~0xdx,which goes to zero as t goes to infinity.As seen from the above equation, the rate of convergence to separable solution is not affected by the asymmetry in cell division. An example of the class of functionsr discussed in Theorem 1 is rt=1/t+k, where k≥1. Also, the analysis in Theorem 1 for rt=1 for all t>0 and α=β=2 recovers the result of Perthame and Ryzhik [4].We now show that if there exists a solution to the problem ((53)) (((2)) and ((3))), then it is unique. The uniqueness of solutions to (1) for constant coefficients was established by Zaidi et al. [5]. Their analysis, however, breaks down for time-dependent coefficients and asymmetric cell division equation. To prove uniqueness, we simplify (53) by applying the transformation
(84)h~x,t=∫x∞ϑε,tdε.This yields(85)∂∂th~x,t+grt∂∂xh~x,t+brth~x,t=brth~αx,t+brth~βx,t.Integrating (34) from 0 to ∞ with respect to x, and using (5) and (3), yields
(86)∂∂th~0,t=brth~0,t,so that
(87)h~0,t=cebRt,for some constant c. Also, equations (2) and (84) give
(88)h~x,0=h~0x=∫x∞ϑ0εdε.Suppose thath~1 and h~2 are two distinct solutions to (85) that satisfy (87) and (88). Let m¯x,t=h~1x,t−h~2x,t. Then, m satisfies
(89)∂∂tm¯x,t+grt∂∂xm¯x,t+brtm¯x,t=brtm¯αx,t+brtm¯βx,tand the conditions
(90)m¯0,t=0,(91)m¯x,0=0.The PDE (89) can be multiplied with sgnm. This gives
(92)∂∂tm¯x,t+grt∂∂xm¯x,t+brtm¯x,t≤brtm¯αx,t+brtm¯βx,t,which, integrating with respect to x from 0 to ∞ and using (5), (90), and (91), yields
(93)ddt∫0∞m¯x,tdx≤0,which shows that ∫0∞∣m¯x,t∣dx is a decreasing function in t, so that
(94)∫0∞m¯x,tdx≤∫0∞m¯x,0dx=0,and consequently,h~1x,t=h~2x,t.We note that ifrt=1 for all t>0, the above analysis establishes the uniqueness of solutions to (4) subject to conditions (2) and (3) for constant coefficients. It also endorses the result of Zaidi et al. [5] for rt≡1 and α=β=2.


## 4. Conclusions

In this paper, we have determined the separable solution to the asymmetric cell division equation (53) subject to conditions (2) and (3) and have shown that solutions to the PDE converge to this separable solution for large time, at least for a certain class of time-dependent coefficients. This longtime behavior is not certain other cases [8].The asymmetry in cell division does not affect the rate of convergence of solutions towards the SSD solution. We have also established the uniqueness of solutions to the problem ((53)) (((2)) and ((3))).The shape of the SSD obtained for time-dependent coefficients is a scaling, in time, of the shape of the SSD solution of the constant coefficients case (see Figure3).Figure 3
The SSD solution obtained by Zaidi et al. [13] for the constant coefficient case and the SSD solution given by the (40) for t=1, α=3, β=3/2, b=1, and g=2 units.We have also determined unique solutions to the symmetric cell division problems (6) and (15) subject to conditions (2) and (3). The large time dynamics of (6) are markedly different from (15). The same time dependence of the growth and division rates plays a crucial role in the convergence of solutions of (6) to the separable solution (2) for large time.



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