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Vladikavkaz Mathematical Journal 2022, Volume 24, Issue 3, P. 78-86
YAK 517.951
DOI 10.46698/p6936-3163-2954-s
ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF DOUBLY DEGENERATE PARABOLIC EQUATIONS WITH INHOMOGENEOUS DENSITY
L. F. Dzagoeva1 and A. F. Tedeev2
1 South Ossetian State University named after A. A. Tibilov, 8 Putin St., Tskhinval 100001, Republic of South Ossetia; 2 Southern Mathematical Institute — the Affiliate of VSC RAS,
53 Vatutina St., Vladikavkaz 362025, Russia E-mail: dzagoeva.larisa74@gmail.com, a_tedeev@yahoo.com
Abstract. In this paper we study the large time behaviour for solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density. Under the suitable assumptions on the data of the problem and on the behaviour of the density at infinity we establish new sharp bound of solutions for a large time. One of the main tool of the proof is new weighted embedding result which is of independent interest. In addition, the proof of uniform estimates of the solution is carried out by modified version of the classical method of De-Giorgi-Ladyzhenskaya-Uraltseva-DiBenedetto. Similar results in the case of power-like density was obtained by one of the author [10]. The approach of this work can be applied for example when studying the qualitative properties of solutions to the Neumann problem for a doubly nonlinear parabolic equation with inhomogeneous density in domains with non-compact boundaries.
Key words: degenerate parabolic equation, inhomogeneous density, weighted embedding, large time behavior.
AMS Subject Classification: 35K92, 35B33, 35E15.
For citation: Dzagoeva, L. F. and Tedeev, A. F. Asymptotic Behavior of the Solution of Doubly Degenerate Parabolic Equations with Inhomogeneous Density, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 78-86. DOI: 10.46698/p6936-3163-2954-s.
1. Introduction
We consider the Cauchy problem for degenerate parabolic equations in the form
p(|x|)ut(x,t) = Am>pu(x,t), (x,t) € St = RN x (0,T), N ^ 1, (1.1)
u(x, 0) = uo(x) ^ 0, a. e. for x € RN. (1.2)
Here
i=1
We assume that p > 1 and p + m — 3 > 0 that is (1.1) is of the slow diffusion type. Additionaly, we assume that p < N. The function p(s) is assumed to be positive nonincreasing, continuous function in [0, œ), satisfying the condition p(0) = 1, and
© 2022 Dzagoeva, L. F. and Tedeev, A. F.
Hi: there exists a positive constant l: l < p, such that the function slp(s) is nonincreasing for all s € (1, to),
H2: there exists a positive constant a < p, such that the function sap(s) is nondecreasing for all s € (1, to).
Note that H1 and H2 imply that there exists y > 1 independent of s, such that Y-1s-a ^ p(s) ^ Ys-1 for all s € (1, to).
The qualitative theory of degenerate parabolic equations with variable coefficients and, in particular, with inhomogeneous density has attracted much attention. This is explained by the fact that the asymptotic properties of solutions essentially depend on the nature of the behavior of the coefficients at infinity. In particular, the typical properties of degenerate equations with constant coefficients, such as the compactness of the support, the behavior at large values of time may not take place depending on the degree of degeneracy of the inhomogeneous density at infinity. Firstly the surprising properties of solutions for degenerate parabolic equations with inhomogeneous density were established in [1]. Namely, in one dimensional Cauchy problem for the general porous media equation (PME) with inhomogeneous density, provided the density decays fast enough the solution of the corresponding Cauchy problem tends to the constant as t — to. The latter means that the solution of the mentioned Cauchy problem as t — to behaves as a solution of the corresponding Neumann problem in a bounded domain. The paper [2] is devoted to the asymptotic behaviour of solutions to the Cauchy problem for inhomogeneous PME for slowly decaying density. These results were extended and developed in [3-7]. Another nonstandard property of degenerate parabolic equation with inhomogeneous density is the possible absence of the finite speed of propagation globally in time see [8]. Results of [8] were generalized in [9, 10] (see also [11]), where in particular, new critical exponents have been found for a doubly degenerate parabolic equations with inhomogeneous density. The paper [12] was devoted to the studying of large time behavior for degenerate Neumann problem in domains with noncompact boundaries. We recommend to the reader interested the qualitative theory of degenerate parabolic equations the survey [13]
The goal of the paper is to obtain the precise rate of stabilization of ||u(t) ||^)Rn as t — to. To this end we need the precise form of the nonpower weighted Sobolev-Gagliardo-Nirenberg which is of independent interest. Here we note that the classical weighted Caffarelli-Kohn-Nirenberg inequality [14] deals with the power-like weights.
Let us start with definition of the weak solution of (1.1), (1.2).
Definition 1.1. By a weak solution of problem (1.1), (1.2) in st we mean a non-negative measurable function u(x,t) such that for a = (p — 1)/(p + m — 2), and any t > 0, u(x,t)1/a belongs to the class Lp((t,T) x RN))nC([t,T]; Li+tJiP(RN)) and (1.1), (1.2) is satisfied in the distributional sense. Moreover, pu(-,t) — puo as t — 0 in Li (Rn). Here, by Wp1(RN) and L1+ap (Rn) we denote the Sobolev space and weighted Lebesgue integral correspondingly.
Before formulating results of the paper, we define
u{s) := p(s)sp, i,p>a{s) := {p~a). (1.3)
Let also
EqAf):=/ P(|x|) If (x)|qdx and Dp(f) := J |V/(x)|pdx.
Jq,p\
rn Rn
Our first result reads as follows.
Theorem 1.1. Let Dp(f), Ea,p(f) < to, where 0 < q ^ p and 0 < a < q, then we have
Eq,p(f)^C(N,p){Dp(f))r
q — q a
w
Ea,pU) (DP(f)fp
q — a p-a
{EaM)YP
P — Q
-a
(1.4)
where ^(-1) is inverse function to
The optimal decay rate is given by the following theorem.
Theorem 1.2. Let u(x, t) be a weak solution of (1.1), (1.2). Assume that ||pu0||Ll(RN) < to, then there exists globaly in time solution u(x, t) in S^ and for any t > 0 the following estimate holds true
... __i
wIIwr") ^ p+m-3
amIKpILraO^
1
p+m — 3
(1.5)
Remark 1.1. Note that if p(s) = (1 + |x|) a, then (1.5) implies that (see [10])
Wl/it) II < 'V II min II (N-a)(P+m-3) + P-a f~ (N-a)(p+m-3) + p-a
llu(t)lLTC(RN) ^ Yllpu°lLi(Rn) t >+P .
Remark 1.2. If a = 0 and m = 1, then the latter coincides with the classical result (see, for example, [15]).
In the proof of Theorem 1.2. we use the classical De Giorgi-Ladyzhenskaya-Uraltseva-Di Benedetto approach in the form of [16, 17]. The rest of the paper is organized as follows. In the Chapter 2 we prove Theorems 1.1 and 1.2. In what follows, we use the symbols 7 > 0, b > 1 for the constants depending on the parameters of the problem, p, m, N only and which may vary from line to line. Moreover, for simplicity we will understand the equation almost everywhere.
2. Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1. < Denote
Eqp(f ):=y |f |qp(|x|) dx, Dp(f ):=y |Vf |pdx.
RN
RN
We have
Eq ,p(f) = J If lq p(|x|) dx + J If |q p(|x|) dx.
Br
By the Holder inequality, we get
J f |qp(|x|) dx ^ I J f r dx
RN\Br \RN
Rn\Br
*
p —a
\f\ap{\x\)—«dx
\
1—
\RN\Br
/
Np
p =
N - p
(2.1)
(2.2)
x
x
p — a
q — a
a
*
p* —a
Making use the S. L. Sobolev inequality:
f |ffdx) < (S(N,p)Dp(f))p,
vrn
where S(N, p) is the sharp constant in this inequality, and owing to the monotonicity of p:
/ P* —a p* —a -,
I/|°p(MK-* dx ^ p(r)p*-i Ea>p(f),
RN\Br
we get from (2.2) that
/ ¿1 i~a q—a p*-q
\f\*p(\x\)dx^(SDp(f)) p p(r)—EaP(f) —
RN\Br
Next, using the classical Hardy inequality
I \f\p\x\~pdx < (1JL-)PDp(f),
N - p
RN
and the Holder inequality, we obtain
(2.3)
If lq p(|x|) dx < If |p |x|-p dx
l/l" \x\pp-i p(\x\) p-1 dx
Br
\RN
Br
q — a r
< C(N,p) ÇDp(f))—« (rpp(r))—« {Ea,p(f))p
q — a — a
P — Q
p — a
Combining now (2.1), (2.3) and (2.4), we get
*
P —Q
* _
Eq,P(f) < (SDp(f)) p p*- Ea>p(f) p'-a
q—a q—a p — Q
+ C(N,p) [Dp{f))—* {rpp{r))—* {EatP(f))—«.
Finally, choosing the free parameter r from the relation:
* —
{SDp(f))^^p(r)f^Ea>p(f)^ =C(N,p)(Dp(f))^ (rpp(r(.Ea,p(f))p
(2.4)
P — Q
—a
we have
EqM)^C1(N,p)(Dp(f))p:
w
_1) f Eqip(f)
{Dp{f))p
q — a p — a
(Ea,P(f))P
P — Q
a
If q = p, we get from the last inequality:
Ep,p(f) < C1 (N, p) Dp(f)
w
Eq,P(f)
q — a
p —q
p — a
p — a
*
a
The proof of (1.4 ) is similar. We give the sketch of the proof only. We have for any R fixed. Applying the Hoolder, Hardy and Sobolev inequalities, we obtain
EPM) = / If lPpdx + f If lppdx < p(R)Rp j If lp Ix—dx
BR RN\BR RN
p — q
/ \ p*-a f ^ p--a
j I ffdx
\RN\BR J
p — a + pp*-a
J IfIapdx
\RN\BR J
/ 1) \p p-a P (P~a) p
where Si = S(N,p)p*(p-a)/(p*-a). Let us choose R from the equality
p(p —a)
R~p^Tp(R) = Dp(f)~p Ea>p(f),
that is
( N — p)(p — a) a
tP(R) := u(R)R 5 = Dp(f)~pEatP(f).
Therefore,
V \Dp{f)p J J
Theorem 1.1 is proved. >
We need the following Caccioppoli type inequality.
Lemma 2.1. Let 9 > 0, and 9 > 2 — m if m < 1, be fixed, and define s = (p + m + 9 — 2)/p. Fix also a1 > a2 > 0, t1 > t2 > 0, r2 > r1. Then
sup / (u(t) — ai)^1 p{\x\) dx + / / \V(u — ai)sApdxdT ^ ———
ri<r<t J + J J \a1 — a2
|m-1|
Br 1 ' 1 Bri
t t (2.5)
x I (ti - T2)-1 J j (u - a2)++1p(|x|) dxdT + (r2 - ri)-p J J (u - a2)+p dxdT I.
^ T1 Br2 T1 Br2 \Br1
For the proof of (2.5) we refer the reader to [16]. Passing r2 ^ to, r1 ^ to in (2.5), we arrive at
sup / (u(t) — a^ + p(|x|) dx + / / \V(u — a1)+| dxdT
1 <T<t j + i i
RN ti rN
|m-1| \ /■
{ti - T2)~l J J (u- a2)e+lp{\x\) dxdT.
(2.6)
Proof of Theorem 1.2.
< Define for h0 > h^ > 0, To > t^ > 0, and i = 0,1,2,...,
p+m + e-2
ki = h00 + (h0 - /¿oo)2-t, U = ^ + (t0 - r00)2"t, fi = (u- ki)+ p
t
Now using the Holder inequality and the embedding (1.4) with f = fi and q = p(1 + d)/(p + m + 9 — 2), a = p/(p + m + 9 — 2) we obtain that
Y
(to - Tc )
fiqP(|x|) dx < Y
RN
(To - Tc)
/f P(M) dx m(T) v
< Ci7 7-r (Dp(fi))p to U™ (cl№(r)
(to - Tc) ^ avDp(fi) + 7\ Cij
2*
(to - Tc)
w ici^¿(t)
№(t) P
I
IM{T) p
(2.7)
Here it is denoted
Pi(t) := J p(|x|) dx.
u(t )>fci
Integrating in time (2.7) and denoting Mi(t) = sup0<T<t pi(T) we get
Y
2*
(to - Tc)
t t J J fq+iP(|x|) dx < e J J |Vfi+i|p dxdT
ti+1 RN v
+ 7£ P-Ï i(r0 - too)
ti+1 RN
ho ho - hc
p| m — 11 p-q
uj U-V (cMl+l(t)^))]p-q Ml+l{t)
Combining now (2.8) and (2.6) with a1 = ki, a2 = ki+1, t} = ti, t2 = ti+1, we get
(2.8)
t t J* := sup I f?p(|x|) dx + / I |Vfi|p dxdT < e [ I |Vf+i|p dxdT
ti<T<t J J J J J
RN
■ — P
+ 7 P-Î i(r0 - too)
ti RN ti+1 Rn
—2L_ / ho
ho - hc
p| m — 11 p-q
q
u U^ (cMl+1(t)^))] p-q Mj_|_i(i).
Iterating this inequality, we get
ho - hc
p|ra — 11
+ p~9HT0-Too) p-«
wi^r^icMooW^)) p~q M^^be)
1
p-q
k=0
Choosing e so small that eb < 1 and letting i — to, we have
sup / /oP(M) dx ^ 7i(ro — Too)
To<T<t
Rn
p|ra — 11
ho \ p-i
ho - hc
2
2
q
p
p
q
p
x
p — q
x
k
x
CO [4 fcMoo^w
1
p-q
M^ (t).
(2.9)
To complete the proof, we need the second iteration. Let k > 0, n = 0,1,2,..., and
Kn = k( 1-2-™"1), Kn = Kn +2Kn+1, tn = t(l-2-n~1). Applying (2.9) with to = t'n+1, r00 = t'n, ho = Kn, hoo = Kn, we deduce from (2.9) that
Yn+1 :=
sup / p(|x|) dx ^ 7bnk-(1+ö)
1<T<t
-1 u(r)>Kn+i
sup / (u — Kn)l^dp(\x\) dx
' <T<t J
(2.10)
tn+1<T<tu(T )>Kn+1
bnk~(-l+d)
uU^lcYr»
i
p-q
Yn, b > 1.
Taking into account the property H1 and H2, which imply that 1)(sA) ^ \N/(N 0^1 1)(s) for 0 < A ^ 1, ^1-1)(sA) ^ 01-1)(s) for A > 1, and Yn ^ Yo, we derive from (2.10) that
1+9
p+m-3 v-av 1+a
Yo Yn ,
a=
N (1 + 9)
(N — l)(p + m — 3)'
Thus the last inequality has a form
Yn+1 < bnCY^+£, C = 7t~pÄ L U[-1) iyc
i+o
P+m-3 £ = a_
Then, using the iterative lemma (see [18, Chap. 2, Lemma 5.6]), we conclude that Yn — 0 as n — to, provided Y0C1/e < b-1/e2, that i
is
_i__±_
k t p+m-3
1
p+m —3
^ S,
(2.11)
where 5 is a sufficiently small constant depending on the data of the problem. Thus u ^ k. Next, note that by the Chebyshev inequality we have
Y0 < sup I p{\x\) dx ^ j sup I u(x, t) p{\x\) dx ^ j [ uop(\x\)dx, 0<T<t J k 0<T<t J k J
u(T)>k u(T)>k RN
where we also have used the estimate |u(t)p|1RN ^ to check that
1 rn (see [10]). Thus, it is enough
_i__±_
k t p+m-3
(k-1||uo p||1,rn) 1/N)
1
p-\-m — 3
^ S.
To this end, we can choose the free parameter k as follows
2__
k = ~t P+m-3
S
wU1 1M (k 1|uOp|1,rn
1/N
1
p+m-3
Then, after taking into account that ||u(t)||^ ^ k and using the monotonicity arguments, we arrive at the desired result. Theorem 1.2 is proved. >
x
x
1
n
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Received October 6, 2021 larisa f. dzagoeva
South Ossetian State University named after A. A. Tibilov, 8 Putin St., Tskhinval 100001, Republic of South Ossetia, Senior Lecturer
E-mail: dzagoeva. larisa74@gmail. com
Anatoli f. Tedeev
Southern Mathematical Institute — the Affiliate of VSC RAS, 53 Vatutina St., Vladikavkaz 362025, Russia, Leading Scientific Researcher E-mail: a_tedeev@yahoo.com https://orcid.org/0000-0001-7883-9795
Владикавказский математический журнал 2022, Том 24, Выпуск 3, С. 78-86
АСИМПТОТИЧЕСКОЕ ПОВЕДЕНИЕ РЕШЕНИЯ ДВАЖДЫ ВЫРОЖДАЮЩИХСЯ ПАРАБОЛИЧЕСКИХ УРАВНЕНИЙ С НЕОДНОРОДНОЙ ПЛОТНОСТЬЮ
Дзагоева Л. Ф.1, Тедеев А. Ф.2
1 Юго-Осетинский государственный университет им. А. А. Тибилова, Республика Южная Осетия, 100001 Цхинвал, ул. Путина, 8; 2 Южный математический институт — филиал ВНЦ РАН,
Россия, 362025, Владикавказ, ул. Ватутина, 53 E-mail: dzagoeva.larisa74@gmail.com, a_tedeev@yahoo.com
Аннотация. В данной работе мы изучаем поведение решений задачи Коши для вырожденных параболических уравнений с неоднородной плотностью при неограниченном возрастании времени. При определенных условиях на параметры задачи и поведения плотностной функции на бесконечности устанавливаются новые точные оценки решений при неограниченном возрастании времени. Одним из основных моментом в доказательстве является новая теорема вложения, представляющая независимый интерес. Кроме того, в доказательстве равномерных оценок решения используется модифицированная версия классического метода Де-Джорджи — Ладыженская — Уральцева — Ди Бенедетто. Аналогичные результаты для неоднородной плотности степенного роста были получены одним из авторов в [10]. Подход данной работы может быть использован также при качественном изучении решений задачи Неймана для дважды нелинейного параболического уравнения в областях с некомпактными границами.
Ключевые слова: вырождающееся параболическое уравнение, неоднородная плотность, весовые вложения, поведение при неограниченном возрастании времени.
AMS Subject Classification: 35K92, 35B33, 35E15.
Образец цитирования: Dzagoeva, L. F. and Tedeev, A. F. Asymptotic Behavior of the Solution of Doubly Degenerate Parabolic Equations with Inhomogeneous Density // Владикавк. мат. журн.—2022.— Т. 24, № 3.—C. 78-86 (in English). DOI: 10.46698/p6936-3163-2954-s.
