The expanding behavior of positive set of a degenerate
parabolic equation
Jiaqing Pan 1
Institute of Mathematics, Jimei University,
Xiamen, 361021, P.R.China
Abstract: This work studies the expanding behavior of positive set of solution and the
continuous dependence on the nonlinearity for a degenerate parabolic partial differential equation ut = ∆φ(u). Our objective is to give an explicit expression of speed of propagation of the
solution and to show that the solution continuously depends on the nonlinearity of the equation.
Keywords : expanding behavior of positive set; continuous dependence on nonlinearity;
Cauchy problem;
AMS(2000) Subject Classifications: 35K10, 35K65,
1
Introduction
Assume u(x, t) be a solution of the Cauchy problem of nonlinear degenerate parabolic equation
½
ut = ∆φ(u)
in Q,
(1.1)
u(x, 0) = u0 (x)
on RN ,
where, Q = RN × R+ and
φ(s) > 0,
φ0 (s) > 0,
φ00 (s) > 0,
φ(0) = φ0 (0) = 0,
(1.2)
and
0 ≤ u0 (x) ≤ M,
0 < ku0 kL1 (RN ) < ∞.
(1.3)
We will consider two questions about the problem (1.1) in this work. The first one we are
interested in is continuous dependence on the nonlinearity of equation (1.1). Exactly, if v(x, t)
is a solution of the Cauchy problem of linear heat equation with the same initial value
½
vt = ∆v
in Q,
(P )
v(x, 0) = u0 (x)
on RN ,
we want to discuss the difference between u and v in Q and give an explicit error-estimate.
In 1981, Ph.Benilan and M.G.Crandall discussed the continuous dependence on φ of (1.1). If
1
Email: [email protected]; [email protected]. This work was supported by the Natural Science Foundation of Fujian province (S0650022).
1
φn : R −→ R is continuous and nondecreasing for all n = 1, 2, 3, ..., φn (0) = 0, they proved that
(see p.162 in [8])
kun − u∞ kL1 (RN ) −→ 0,
as φn −→ φ∞ .
Where un are the solutions of the Cauchy problem
½
ut = ∆(φn (u))
u(x, 0) = u0 (x)
in Q,
on RN .
However, as pointed by [1], the result of [8] are not written in terms of explicit estimates. To
study the problem more precisely, B.Cockburn and G.Gripenberg (see [1]), in 1999, extended
the result of [8] for the Cauchy problem of degenerate parabolic equation
½
vt = ∆(φ(v)) + ∇ · (Φ(v))
in Q,
(1.4)
v(0) = h
on RN
with the conditions
Φj ∈ C 1 (R, RN ), Φj (0) = 0 and, φj (0) = 0, φ0j (t) > 0, t > 0
for j = 1, 2. Their explicit estimate obtained in [1] is
kv1 (·, t) − v2 (·, t)kL1 (RN ) ≤ kh1 − h2 kL1 (RN ) + kh1 kT V (RN )
Ã
×
!
q
q
t · sup kΦ01 (s) − Φ02 (s)kL∞ (RN ) + 4 tN sup | φ01 (s) − φ02 (s)|
(1.5)
√
s∈I(h1 )
s∈I(h1 )
for any t > 0. Where I(h) = (−kh− k∞ , kh+ k∞ ). Clearly, (1.5) tells us that the error-difference
kv1 (·, t) − v2 (·, t)kL1 (RN ) is not necessary bounded as t −→ ∞. However, if we set φ(v) =
v m (m > 1) and Φ = 0 in (1.4), the conservation of total mass (see [5]) claims
kvi (·, t)kL1 (RN ) = khi kL1 (RN )
i = 1, 2
for all t > 0. Therefore, the error-difference is bounded with respect to t > 0 uniformly and in
fact,
kv1 (·, t) − v2 (·, t)kL1 (RN ) ≤ kh1 kL1 (RN ) + kh2 kL1 (RN )
t > 0.
(1.6)
Comparing (1.5) and (1.6), we see that the result (1.5) may be improved. Thus we will discuss
the question more precisely and give an explicit error-difference formula. If u is a solution of
(1.1), then the first result of our work is
1
1
kv − ukL2 (QT ) ≤ C · max |s j − (φ(s)) j |
s∈[0,M ]
2
for large T > 0, with C = O(T γ ) and γ = 1 −
√
(x−ξ)2
N R
(P ), v = (2 πt)− 2 RN u0 (ξ)e− 4t dξ.
(j−2)N
j(2+N )m , j
= 2, 3, 4, .... Where v is the solution of
Denote positive set of u by
Hu (t) = {x ∈ RN : u(x, t) > 0}
t > 0.
The second aim of this paper is to discuss the expanding behavior of Hu (t). It is well-known
that the study on this topic has a long history. Clearly, if we only consider uniformly parabolic
equation, for example, the linear heat equation
ut = ∆u,
we see that u(x, t) > 0 in Q if only its initial value u0 satisfies (1.3). That is to say, the speed of
propagation of u is infinite in this case. However, this fact is not true for degenerate parabolic
equation. For example, L.A.Peletier and B.H. Gilding (see [2, 6], ) discussed the free boundary
of degenerate parabolic equations
∂u
∂ 2 φ(u)
=
∂t
∂x2
and
∂u
∂ 2 um ∂un
=
+
∂t
∂x2
∂x
respectively. They proved that the speeds of propagation of the solutions are finite. But they
got no explicit formulas. As to the case of the dimension N ≥ 1, Barenblatt (see [4]) discussed
the problem
½
ut = ∆(um )
in Q,
(1.7)
u(x, 0) = δ(x)
on RN ,
and got a source type solution
B(x, t, C) = t
−λ
¸ 1
·
|x|2 m−1
C − κ 2µ
,
t
+
(1.8)
where, m > 1, [h]+ = max{h, 0} and
λ=
N
,
N (m − 1) + 2
µ=
λ
,
N
κ=
λ(m − 1)
.
2mN
Clearly, (1.8) says
(
HB (t) =
r
x ∈ RN : |x| <
C µ
t
k
)
.
Thus, we see the exactly expanding behavior of HB (t):
r
C µ
|x| =
t
x ∈ ∂HB (t).
k
3
(1.9)
To extend the result of [4], J. L. Vazquez (see [5]) proved that the solution of the Cauchy
problem of the equation ut = ∆(um ) (m > 1) with a general initial u0 (x) satisfying (1.3) also
has a bounded positive set Hu (t):
c1 tµ ≤ |x| ≤ c2 tµ
x ∈ ∂Hu (t).
where c1 , c2 are positive constants. Hence, the speed of propagation of Hu (t) is the similar as
one of HB (t). However, for the general parabolic equation in (1.1) with the dimension N ≥ 1,
there are no else explicit results on the speed of propagation up to now. Thus, the present work
will investigate this question and gives the second result:
Ã
!2
sup |x|
x∈Hu (t)
≥ 4C0 t
for all t > 0.
Where u is the solution of (1.1), C0 is determined later.
Without loss of generality, we suppose the function φ satisfies (1.2) and
φ00 (s)φ(s)
≥ C0
(φ0 (s))2
for s ≥ 0.
(1.10)
for some given positive constant C0 . We note that the condition (1.10) is not strict because we
00
m−1
have the example φ(s) = sm (m > 1). For this case, φ(φ(s)φ(s)
0 (s))2 =
m , so (1.10) follows.
Besides the two conclusions mentioned above, this paper intends to prove some useful estimates. For example, it is well-known (see [3, 5]) that if u(x, t) is a solution of the Cauchy
problem of the equation ut = ∆(um ) (m > 1), then
∂u
λu
≥− ,
∂t
t
(1.11)
where λ is defined by (1.9). This inequality is very important for the study of equation ut =
∆(um ), many interesting results can be resulted from it. But for the general parabolic equation
(1.1), few such estimates can be found. So we will establish a similar one:
∂φ(u)
φ(u)
≥−
,
∂t
C0 t
(1.12)
where C0 is defined by (1.10). In addition, to calculate the expanding behavior of Hu (t) accurately, we will give a more precise Poincaré inequality.
2
The main results
Lemma 2.1 (existence and uniqueness ) The Cauchy problem (1.1) has an unique
positive solution u(x, t) if (1.2), (1.3) are satisfied. Moreover,
4
(i) conservation of total mass:
Z
Z
u(x, t)dx =
RN
RN
(ii) L1 -order-contraction property:
Z
Z
[u1 (x, t2 ) − u2 (x, t2 )]+ dx ≤
RN
RN
u0 (x)dx
t > 0;
[u1 (x, t1 ) − u2 (x, t1 )]+ dx
(2.1)
0 ≤ t1 < t 2 ,
(2.2)
where u1 , u2 are the two solutions of (1.1) with the initial data u10 and u20 respectively, where
½
u1 − u2
u1 − u2 ≥ 0,
[u1 − u2 ]+ =
¤
0
u1 − u2 < 0.
Lemma 2.2 Let u(x, t) be the solution of problem (1.1) with (1.2), (1.3) and (1.10). Then
φ(u)
∂φ(u)
≥ −
∂t
C0 t
in D0 (Q).
(2.3)
Lemma 2.3 Let u be the solution of the Cauchy problem (1.1) with (1.2), (1.3) and (1.10).
Then there is a C2 > 0 such that
φ(u) ≤ C2 t
N
− (2+N
)
in Q.
(2.4)
Where C2 depends on N , M and ku0 kL1 (RN ) .
Theorem 2.1 Assume
φ(s) ≥ C · sm
for some C > 0 and m > 1, u be the solution of (1.1) with (1.2), (1.3) and (1.10). For every
given positive integer number j ≥ 2, there is a C3 > 0 such that
1
1
kv − ukL2 (QT ) ≤ C3 · max |s j − (φ(s)) j |
s∈[0,M ]
for large T . Where, C3 = O(T γ ) with
γ =1−
(j − 2)N
,
j(2 + N )m
v is solution of the Cauchy problem
½
vt = ∆v
v(x, 0) = u0 (x)
in Q,
on RN .
5
(2.5)
Theorem 2.2 Let u0 satisfy (1.3) and supp u0 = Bε with ε > 0, u is the solution of the
Cauchy problem (1.1) with (1.2), (1.3) and (1.10). Then for every t > 0,
√
C0 t
|xt | ≥
,
(2.6)
2
where C0 is defined by (1.10) and |xt | = sup |x|. Specially, |xt | does not depends on ε.
x∈Hu (t)
References
[1] B.Cockburn & G.Gripenberg, Continuous dependence on the nonlinearities of solutions of
degenerate parabolic equations J.Differential Equations, 151 (1999), 231-251
[2] B.H. Gilding, Properties of solutions of an equation in the theory of infiltration, Arch.
Rational Mech. Anal. , 65 (1977), 203-225
[3] D.G.Aronson,Ph.Benilan, Régularité des solutions de l’équation des milieux poreux dans
RN ,C.R.Acad.Sci.Paris Sér.A. , 288(1979), 103-105
[4] G.I.Barenblatt, On some unsteady motions and a liquid or a gas in a porous medium, Prikl.
Mat. Mech. 16(1952), 67-78 (in Russian)
[5] J. L. Vazquez, An introduction to the mathematical theory of the porous medium equation,
in “Shape Optimization and Free Boundaries” (M. C. Delfour and G. Sabidussi, eds.),
Kluwer Academic, Dordrecht, (1992), 347-389
[6] L.A.Peletier, On the Existence of an Interface in Nonlinear Diffusion Processes, Lecture
Notes in Mathematics, Berlin: Springer, 415(1974), 412–416
[7] O.A.Ladyshenskaya,V.A.Solounikov and N.N.Uraltseva, Linear and quasilinear equations
of parabolic type, Am.Math.Soc.Providence, R.I, 1968
[8] Ph.Benilan, M.G.Crandall, The continuous dependence on φ of solution of ut − M φ(u) = 0,
Indiana Univ. Math. J. 30(1981), 161-177
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