Asymptotic Almost Periodicity of Scalar Parabolic Equations with Almost Periodic Time ...

26 Pages·2010·0.19 MB·English

Asymptotic Almost Periodicity of Scalar Parabolic Equations
with Almost Periodic Time Dependence
Wenxian Shen
Department of Mathematics
Auburn University
Auburn University, AL 36849
and
Yingfei Yi *
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332
* Partially supported by NSF grant DMS 9207069
1
1. Introduction
This paper is devoted to the study of asymptotic almost periodicity of bounded solu-
tions for the following time almost periodic one dimensional scalar parabolic equation:

u = u +f(t,x,u,u ), t > 0, 0 < x < 1,
t xx x
(1.1)

u(t,0) = u(t,1) = 0, t > 0,
where f : IR1×[0,1]×IR1×IR1 → IR1 is a C2 function, and f(t,x,u,p) with all its partial
derivatives (up to order 2) are (Bohr) almost periodic in t uniformly for other variables in
compact subsets.
Denote by H(f) the hull of f in compact open topology and let Xα be a fractional
power space associated with the operator u → −u : H2(0,1) → L2(0,1) that satisfies
xx 0
Xα (cid:44)→ C1[0,1] (that is, Xα is compact embedded in C1[0,1]). Equation (1.1) generates a
(local) skew product semiflow Π on Xα ×H(f) (see section 2) as follows:
t
Π (U,g) = (u(t,·,U,g),g ·t), (1.2)
t
where g · t is the flow on H(f) defined by time translations (H(f) is therefore almost
periodic minimal under this flow, see [9], [20]), u(t,·,U,g) is the solution of

u = u +g(t,x,u,u ), t > 0, 0 < x < 1,
t xx x
(1.3)
g

u(t,0) = u(t,1) = 0, t > 0
with u(0,·,U,g) = U(·).
We shall study the asymptotic almost periodicity for a positively bounded motion
Π (U,g)of(1.2)byinvestigatingitsω-limit setω(U,g)(thesetofallaccumulationpointsof
t
Π (U,g) as t goes to infinite) since it has been shown in [22] that Π (U,g) is asymptotically
t t
almost periodic if and only if ω(U,g) is an almost periodic extension of H(f) (namely, a
1-cover of H(f)). We note that for (1.2), an ω-limit set is an almost periodic extension of
H(f) if and only if it is almost periodic minimal ([22]).
For a time periodic parabolic equation of form (1.1), it is known that any bounded
solution is asymptotically periodic, that is, eachω-limit set of (1.2) is necessary a 1-cover of
H(f) ∼ S1 (see [2], [5]). Nevertheless, similar results are false for the time almost periodic
parabolic equation (1.1) since an ω-limit set of (1.2) may not be even minimal (see [18]
2
and an example in section 5 of the current paper). In general, following the results of [22],
even an ω-limit set of (1.2) is minimal, one expects that rather than an almost periodic
extension it may be an almost automorphic extension (namely, an almost 1-cover of H(f))
or a proximal extension of H(f) (proximal means that two trajectories of (1.2) starting
on a same fibre clasp eventually following a time sequence). However, similar to [18] for
the scalar ODE case, it is shown in [22] that if an ω-limit set of (1.2) is uniformly stable,
then it is almost periodic minimal (see also [21], [23], [24] for different situations of almost
periodic parabolic equations).
In this paper, we shall show that the hyperbolicity of ω(U,g) also implies its almost
periodicity. More precisely, we have the following main results:
1) Let ω(U ,g ) ⊂ Xα × H(f) be an ω-limit set of (1.2). Suppose that ω(U ,g ) is
0 0 0 0
hyperbolic, that is, the linearized equation about the flow on ω(U ,g ) has an ED
0 0
(exponential dichotomy) on ω(U ,g ) and the projections P(y) (y ∈ ω(U ,g )) associ-
0 0 0 0
ated to the ED satisfy ImP(y) (cid:54)= {0} for y ∈ ω(U ,g ) (see section 2). Then ω(U ,g )
0 0 0 0
is an almost periodic extension of H(f), that is, Π (U ,g ) is asymptotically almost
t 0 0
periodic.
2) Suppose that f(t,x,u,p) = F(ω t,ω t,···,ω t,x,u,p) is quasi-periodic in t and F :
1 2 k
Tk × [0,1] × IR2 → IR1 is Cr,γ (r ≥ 2, 0 < γ ≤ 1) (that is, F and all its partial
derivatives up to order r are locally H¨older continuous with H¨older exponent γ if
0 < γ < 1, and are locally Lipschitz continuous if γ = 1). Then a hyperbolic ω-limit
set ω(U ,g ) is Cr,γ diffeomorphic to Tk and the flow on ω(U ,g ) is Cr,γ conjugate
0 0 0 0
to the twist flow on Tk.
We remark that the above results are false for higher dimensional parabolic equations.
There are examples even in autonomous two dimensional parabolic equations in which a
hyperbolic invariant set may be rather chaotic ([17]). We also remark that if the flow on
an ω-limit set ω(U ,g ) of (1.2) has an ED but the projections P(y) associated to the ED
0 0
satisfy ImP(y) ≡ {0} for y ∈ ω(U ,g ), then ω(U ,g ) is uniformly stable. The above
0 0 0 0
result 1) in this case then follows from [22] and the above result 2) follows from arguments
in Theorem 4.9 of the current paper. Furthermore, we note that for a time almost periodic
scalarODE,similarresultsasabovetriviallyholdsincethehyperbolicityinthescalarODE
case is equivalent to either strong stability or strong instability. Following the arguments
3
of the current paper and the Floquet theory ([7]), our main results also hold true in the
case of Neumann boundary conditions.
The paper is organized as follows. In section 2, we summarize some of the preliminary
materials such as the zero number property from [1], [16], invariant manifold theory due to
[6], [11], [25] and Floquet theory developed in [7]. We also sketch the construction for the
skew product semiflow (1.2). We discuss the zero crossing numbers on invariant manifolds
in section 3 similar to [3], [4]. The main results are proved in section 4. An example with
a nonhyperbolic ω-limit set is described in section 5.
Acknowledgment:Wewouldliketothanktherefereeforcommentsandcarefulreading
of the current paper.
2. Preliminary
In this paper, the norm symbol (cid:107) · (cid:107) will have its obvious meaning unless specified
otherwise.
1. Exponential Dichotomy (ED in short).
Consider
(cid:48)
v = A(y ·t)v, t > 0, y ∈ Y, v ∈ X, (2.1)
where Y is a compact metric space, y · t is a flow in Y, X is a Banach space, A(y) :
D(A(y)) → X is a linear operator, here X is a Banach space and D(A(y)) (cid:44)→ X (cid:44)→ X .
0 0 0
Suppose that the evolution operator Φ(t,y) : X → X of (2.1) (t ≥ 0, y ∈ Y) exists in
the usually sense, that is, Φ(0,y) = I (the identity), and Φ(t,y)v ∈ D(A(y·t)), Φ(t,y)v is
differentiable in t with respect X norm and satisfies (2.1) for t > 0, moreover
0
Φ(t+s,y) = Φ(t,y ·s)Φ(s,y), t,s ∈ IR+,y ∈ Y. (2.2)
Definition 2.1. Equation (2.1) is said to have an exponential dichotomy on Y if there
exist β > 0, K > 0 and continuous projections P(y) : X → X such that for any y ∈ Y,
the following hold :
1) Φ(t,y)P(y) = P(y ·t)Φ(t,y), t ∈ IR+;
2) Φ(t,y)| : R(P(y)) → R(P(y · t)) is an isomorphism for t ∈ IR+ (hence
R(P(y))
Φ(−t,y) := Φ−1(t,y ·−t) : R(P(y)) → R(P(y ·−t)) is well defined for t ∈ IR+);
4
3)
(cid:107)Φ(t,y)(I −P(y))(cid:107) ≤ Ke−βt, t ∈ IR+,
(2.3)
(cid:107)Φ(t,y)P(y)(cid:107) ≤ Keβt, t ∈ IR−.
Remark 2.1. 1) (2.3) is equivalent to
(cid:107)Φ(t−s,y ·s)(I −P(y ·s))(cid:107) ≤ Ke−β(t−s), t ≥ s, t,s ∈ IR1,
(2.4)
(cid:107)Φ(t−s,y ·s)P(y ·s)(cid:107) ≤ Keβ(t−s), t ≤ s, t,s ∈ IR1
for any y ∈ Y.
2)
R(P(y)) = {v ∈ X|Φ(t,y)v exists for t ∈ IR1,
Φ(t,y)v → 0 exponentially as t → −∞}
= {v ∈ X|Φ(t,y)v exists for t ∈ IR1, Φ(t,y)v → 0 as t → −∞},
and
R(I −P(y)) = {v ∈ X|Φ(t,y)v → 0 exponentially as t → ∞}
= {v ∈ X|Φ(t,y)v → 0 as t → ∞}.
Definition 2.2. Vs(y) := R(I−P(y)) and Vu(y) := R(P(y)) are referred to as the stable
and unstable subspaces of (2.1) at y ∈ Y.
2. Stable and Unstable Manifolds.
Consider
(cid:48)
v = A(y ·t)v +F(v,y ·t), t > 0, y ∈ Y, v ∈ X, (2.5)
where F(·,y) ∈ C1(X,X ), F(v,·) ∈ C0(Y,X ) (v ∈ X), F(v,y) = o((cid:107)v(cid:107)), and A, X, X ,
0 0 0
Y, y · t are as in (2.1). We assume that the solution operator Λ (·,y) of (2.5) exists in
t
usual sense (that is, Λ (v,y) = v, Λ (v,y) ∈ D(A(y·t)) , Λ (v,y) is differentiable in t with
0 t t
respect to X norm and satisfies (2.5) for t > 0).
0
Theorem 2.1. Consider (2.5) and assume that
1) Equation (2.1) has an ED on Y with ED constants K, β (that is, (2.4) holds);
2) The evolution operator Φ(t,y) : X → X of (2.1) can be extended to X , and
0
Φ(t,y)X ⊂ X for t > 0;
0
(cid:101) (cid:101)
3) There are constants ρ,K with 0 ≤ ρ < 1,K > 0 such that
5
(cid:107)Φ(t−s,y ·s)(I −P(y ·s))v(cid:107) ≤ K(cid:101)e−β(t−s)(t−s)−ρ(cid:107)v(cid:107) , t > s, t,s ∈ IR1,
X X
0
(2.6)
(cid:107)Φ(t−s,y ·s)P(y ·s)v(cid:107) ≤ K(cid:101)eβ(t−s)(cid:107)v(cid:107) , t ≤ s, t,s ∈ IR1
X X
0
for any v ∈ X and y ∈ Y, where Φ, P are as in (2.4).
0
Then, (2.5) possesses for each y ∈ Y a local stable manifold Ws(y) and a
local unstable manifold Wu(y) which satisfy the following properties:
1) There are δ∗ > 0, M > 0, and bounded continuous functions hs,u : ∪ Vs,u(y) ×
y∈Y
{y} → ∪ Vu,s(y) with hs,u(·,y) : Vs,u(y) → Vu,s(y) being C1 for each fixed y ∈ Y,
y∈Y
∂hs,u
and hs,u(v,y) = o((cid:107)v(cid:107)), (cid:107) (v,y)(cid:107) ≤ M for all y ∈ Y, v ∈ Vs,u(y) such that
∂v
(cid:169) (cid:170)
Ws,u(y) = vs,u +hs,u(vs,u,y)|vs,u ∈ Vs,u(y)∩{v ∈ X|(cid:107)v(cid:107) < δ∗} .
0 0 0
Moreover, Ws,u(y) are diffeomorphic to Vs,u(y)∩{v ∈ X|(cid:107)v(cid:107) < δ∗}, and Ws,u(y) are
tangent to Vs,u(y) at 0 ∈ X for each y ∈ Y.
2) Ws(y) and Wu(y) are locally invariant in the sense that for v ∈ Ws,u(y) there are
s,u
intervals Is = [0,τ), Iu = (−τ,0] for some τ > 0 such that Λ (v ,y) ∈ Ws,u(y ·t)
t s,u
for t ∈ Is,u. They are also overflowing invariant in the sense that
Λ (Ws(y),y) ⊂ Ws(y ·t) for t (cid:192) 1,
t
Λ (Wu(y),y) ⊂ Wu(y ·t) for t (cid:191) −1.
t
3) There are δ∗,δ∗ > 0 such that
1 2
{v ∈ X|Λ (v,y) → 0 as t → ∞,(cid:107)Λ (v,y)(cid:107) < δ∗ for all t ≥ 0} ⊂ Ws(y)
t t 1
⊂ {v ∈ X|(cid:107)v(cid:107) < δ∗,Λ (v,y) → 0 as t → ∞},
2 t
and
{v ∈ X|Λ (v,y) → 0 as t → −∞,(cid:107)Λ (v,y)(cid:107) < δ∗ for all t ≤ 0} ⊂ Wu(y)
t t 1
⊂ {v|(cid:107)v(cid:107) < δ∗,Λ (v,y) → 0 as t → −∞}.
2 t
4) There is a constant C > 0 such that for any y ∈ Y and v ∈ Ws(y), v ∈ Wu(y), one
s u
has
(cid:107)Λt(vs,y)(cid:107) ≤ Ce−β2t(cid:107)vs(cid:107) for t ≥ 0,
(cid:107)Λt(vu,y)(cid:107) ≤ Ceβ2t(cid:107)vu(cid:107) for t ≤ 0.
6
The proof of the theorem follows from the arguments in [6], [11], [25].
Remark 2.2. 1) By 3), if v ∈ X is such that Λ (v,y) → 0 as t → ∞ (−∞), then
t
there is a T > 0 such that Λ (v,y) ∈ Ms(y·t) for t ≥ T (Λ (v,y) ∈ Mu(y·t) for t ≤ −T).
t t
2) For any y ∈ Y, v ∈ Ws(y), and v ∈ Wu(y), let vs(v ,y) = (I − P(y))v and
s u 0 s s
vu(v ,y) = P(y)v . Then,
0 u u
Λ (v ,y) = vs(Λ (v ,y),y ·t)+hs(vs(Λ (v ,y),y ·t),y ·t) for t (cid:192) 1,
t s 0 t s 0 t s
(2.7)
1
Λ (v ,y) = vu(Λ (v ,y),y ·t)+hu(vu(Λ (v ,y),y ·t),y ·t) for t (cid:191) −1,
t u 0 t u 0 t u
and
(cid:107)v0s(Λt(vs,y),y ·t)(cid:107) ≤ C(cid:101)e−β2t(cid:107)vs(cid:107) for t ≥ 0,
(2.7)
2
(cid:107)v0u(Λt(vu,y),y ·t)(cid:107) ≤ C(cid:101)eβ2t(cid:107)vu(cid:107) for t ≤ 0,
(cid:101)
where C is a positive constant which is independent of y, v , and v .
s u
3) If A(y) in (2.5) is of form A + B(y), where −A is a sectional operator in X ,
0 0 0
B(y) : X → X is uniformly bounded and B(y ·t) is locally H¨older continuous in t ∈ IR1
0
for any y ∈ Y, and if X (X ⊃ D(A ), X (cid:54)= D(A )) is a fractional power space of −A ,
0 0 0
then the conditions 2), 3) in Theorem 2.1 are consequences of the condition 1) (see Lemma
7.6.2 of [11]).
4) Suppose that A(·) is as in 3). If F(v,y ·t) is locally H¨older continuous in t ∈ IR1
(cid:101)
for any v ∈ X, y ∈ Y, then the solution operator Λ (·,y) of (2.5) exists ([11]). Let X be a
t
(cid:101)
fractional power space of −A with X (cid:44)→ X (cid:44)→ X . Assume that F(·,·) can be extended
0 0
to X(cid:101) × Y with F(·,y) ∈ C1(X(cid:101),X ), F(v,y) = o((cid:107)v(cid:107) ), and F(v,y · t) is locally H¨older
0 (cid:101)
X
continuous in t ∈ IR1 for any v ∈ X(cid:101), y ∈ Y. Then hs(·,y) and hu(·,y) in the theorem can
be extended to (I −P(y))X(cid:101) and P(y)X(cid:101) with hs,u(v,y) = o((cid:107)v(cid:107) ).
(cid:101)
X
3. Zero Number Properties.
For a given C1 function v : [0,1] → IR1, the zero number of v is defined as
Z(v(·)) = #{x ∈ (0,1)|v(x) = 0}.
The following properties can be found in [1] and [16].
Lemma 2.2. Consider the scalar linear parabolic equation:

v = a(t,x)v +b(t,x)v +c(t,x)v, t > 0, x ∈ (0,1),
t xx x
(2.8)

v(t,0) = v(t,1) = 0, t > 0,
7
where a, a , a , a , b, b , b and c are bounded continuous functions, a ≥ δ > 0. Let
t x xx t x
v(t,x) be a classical nontrivial solution of (2.8). Then, the following hold:
1) Z(v(t,·)) is finite for t > 0 and is nonincreasing as t increases;
2) Z(v(t,·)) can drop only at t such that v(t ,·) has a multiple zero in [0,1];
0 0
3) Z(v(t,·)) can drop only finite times, and there exists a t∗ > 0 such that v(t,·) has only
simple zeros in [0,1] as t ≥ t∗ (hence Z(v(t,·)) = constant as t > t∗).
4. Floquet Theory.
Consider the following linear parabolic equation:

w = w +b(x,y ·t)w, t > 0, 0 < x < 1,
t xx
(2.9)

w(t,0) = w(t,1) = 0, t > 0,
where y·t is a flow on a compact metric space Y, b : [0,1]×Y → IR1 is continuous. Suppose
that for any w ∈ L2(0,1), the solution w(t,x,w ,y) of (2.9) with w(0,x,w ,y) = w (x)
0 0 0 0
exists. The following results are due to [7].
Theorem 2.3. 1) There is a sequence {w }∞ , w : [0,1] × Y → IR1 (n = 1,2,···)
n n=1 n
such that w (·,y) ∈ C1,γ[0,1] for any γ with 0 ≤ γ < 1, w (0,y) = w (1,y) = 0, and
n n n
(cid:107)w (·,y)(cid:107) = 1 for any y ∈ Y. {w (·,y)}∞ forms a (Floquet) basis of L2(0,1)
n L2(0,1) n n=1
and Z(w (·,y)) = n − 1 for all y ∈ Y. Let W (y) = span{w (·,y)}, n = 1,2,···.
n n n
(cid:76)
Then n2 W (y) = {w ∈ L2(0,1)|w(t,·,w ,y) is exponentially bounded in L2(0,1), and
i=n i 0 0
1
n −1 ≤ Z(w(t,·,w ,y)) ≤ n −1 for all t ∈ IR1}∪{0} for any n , n with n ≤ n .
1 0 2 1 2 1 2
(cid:80)
2) Suppose w (x) = ∞ c0w (x,y) (c0’s are called Fourier coefficients). Then
0 n=1 n n n
(cid:88)∞
w(t,x,w ,y) = c (t)w (x,y ·t), (2.10)
0 n n
n=1
where
(cid:48)
c = µ (y ·t)c , (2.11)
n n n
(cid:82)
c (0) = c0, µ (y ·t) = 1[b(x,y ·t)w (x,y ·t)2 −w (x,y ·t)2]dx, n = 1,2···. Moreover,
n n n 0 n nx
for each n ≥ 1, there are T > 0, κ > 0 which are independent of y ∈ Y such that
n n
(cid:90) (cid:90)
t+T t+T
n n
µ (y ·s)ds− µ (y ·s)ds ≤ −κ , (2.12)
n+1 n n
t t
for all y ∈ Y and t ∈ IR1.
8
3) Define Ψ(·) : Y → L(L2(0,1),l2) by Ψ(y)w = {c0}∞ , where w (x) =
0 n n=1 0
(cid:80)
∞ c0w (x,y). Then Ψ(y · t)w(t,x,w ,y) = {c (t)}, here c (t)’s are given in (2.10).
n=1 n n 0 n n
Moreover, Ψ is continuous, Ψ(y) is an isomorphsim for each y ∈ Y, and there are positive
constants K , K which are independent of y such that
1 2
(cid:107)Ψ(y)(cid:107) ≤ K and (cid:107)Ψ−1(y)(cid:107) ≤ K .
1 2
5. Skew Product Flow.
Consider 
u = u +f(t,x,u,u ), t > 0, 0 < x < 1,
t xx x
(2.13)

u(t,0) = u(t,1) = 0, t > 0,
where f : IR1 ×[0,1]×IR1 ×IR1 → IR1 is a C2 function and f(t,x,u,p) as well as all its
partial derivatives up to order 2 are (Bohr) almost periodic in t uniformly for (x,u,p) in
compact sets of [0,1]×IR1 ×IR1.
Let C := C(IR1 × [0,1] × IR1 × IR1,IR1) be the space of continuous functions P :
IR1 × [0,1] × IR1 × IR1 → IR1. Give C the compact open topology. This topology is
metrizable. To be more precise, for any P,Q ∈ C, define
(cid:88)∞
min{1,(cid:107)P −Q(cid:107) }
n
d(P,Q) = ,
2n
n=1
where
(cid:107)P −Q(cid:107) = sup |P(t,x,u,p)−Q(t,x,u,p)|.
n
t∈IR1,x∈[0,1],|u|≤n,|p|≤n
Then (C,d) is a metric space and the time translation (P,t) → P · t, P · t(s,x,u,p) =
P(t + s,x,u,p) defines a flow on C ([20]). Let H(f) = cl{f · t|t ∈ IR1} be the hull of
f. Then H(f) ⊂ C is an almost periodic minimal set under the translation flow (see
[20]). Define F : H(f) × [0,1] × IR1 × IR1 → IR1 by F(g,x,u,p) := g(0,x,u,p). Then
F(g ·t,x,u,p) ≡ g(t,x,u,p). Since each g ∈ H(f) is C2 ([12]), F(g ·t,x,u,p) is C2 in t,
x, u and p. Moreover, for (g ,x,u,p) ∈ H(f)×[0,1]×IR2 (i = 1,2), it is easy to see that
i
9
|F(g ,x,u,p)−F(g ,x,u,p)| ≤ d(g ,g ), and d(g ·t,g ·t) ≤ d(g ,g ) ([25]). By the above
1 2 1 2 1 2 1 2
construction, equation (2.13) gives rise to a family of equations on H(f):

u = u +F(g ·t,x,u,u ), t > 0, 0 < x < 1,
t xx x
(2.14)
g

u(t,0) = u(t,1) = 0, t > 0,
where g ∈ H(f). Note that (2.14) coincides with (2.13).
f
Let Xα be a fractional power space associated with the operator A : H2(0,1) →
0 0
X ≡ L2(0,1): A u = −u , that satisfies Xα (cid:44)→ C1[0,1], Xα ⊃ D(A ), Xα (cid:54)= D(A ).
0 0 xx 0 0
Define F(cid:101) : Xα ×H(f) → X by F(cid:101)(u,g)(x) = F(g,x,u,u ). Then F˜ is Lipschitz in g and
0 x
C2 in u ([11]), and (2.14) gives rise to an ODE in Xα:
g
u(cid:48) = A u+F(cid:101)(u,g ·t), u ∈ Xα, g ∈ H(f). (2.15)
0
Let u(t,x,U ,g) (u(t,U ,g)) be the solution of (2.14) ((2.15)) with u(0,x,U ,g) =
0 0 g 0
U (x) (u(0,U ,g) = U ∈ Xα). This solution locally exists and also continuously depends
0 0 0
on U ∈ Xα and g ∈ H(f) ([11]). Thus, (2.14) or (2.15) generates a (local) skew product
0 g
semiflow Π on Xα ×H(f):
t
Π (U ,g) := (U ,g)·t := (u(t,U ,g),g ·t) = (u(t,·,U ,g),g ·t),t > 0. (2.16)
t 0 0 0 0
Moreover, following [11] and the standard a priori estimates for parabolic equations, if a
motion (U,g)·t is bounded for t in its existence interval, then it is globally defined, and for
any δ > 0, {(U,g)·t|t ≥ δ > 0} is relatively compact both in Xα×H(f) and in H2(0,1)×
H(f), hence, the ω-limit set ω(U,g)| = ω(U,g)| . Furthermore, u has
Xα×H(f) H2(0,1)×H(f)
continuous derivatives u , u in (0,∞)×[0,1]. Note also that the skew product semiflow
tx xxx
(2.15) has a unique continuous backward extension on ω(U,g)([10]), that is, it is in fact a
usual skew product flow on ω(U,g).
Now, letY ⊂ Xα×H(f)beacompactinvariantsetof(2.16). Foreachy = (U,g) ∈ Y,
the linearized equation of (2.14) about y ·t = (U,g)·t reads:
g

v = v +a(x,y ·t)v +b(x,y ·t)v, t > 0, 0 < x < 1,
t xx x
(2.17)

v(t,0) = v(t,1) = 0, t > 0,
10