The support behavior of the solution to the Cauchy problem for higher order weighted parabolic equations

We consider the Cauchy problem in St = RN x (0, T) for the higher order degenerate parabolic equations of the form d flulq—2u^

f °     d t    + ( — Dm E D ^a (f (( | x | ) | D m u | p D^)) = 0,         (1.1)

| α | = m

u(x, 0) = uo(x),   x G RN.                               (1.2)

• ^# The research was executed at North-Caucasus Center for Mathematical Research of the Vladikavkaz Scientic Center of the Russian Academy of Sciences with the support of the Ministry of Science and Higher Education of the Russian Federation, agreement № 075-02-2024-1379.

(0 2024 Kasaeva, A. R. and Tedeev, A. F.

Here the sum extends to all x-derivatives of order m, iDmul = ^|a|=m|Da u|, f (s) = exp(g(s)), where g(s) be a smooth enough increasing function defined in (0, от). The problem under consideration is doubly nonlinear for p = 2 and q = 2. In what follows, we need to specifier the behavior of g(s) as follows. Following [1, 2], set h E (C 1RN\{xo}))N be satisfying (x — xo) • h ^ 0 with some xo E RN. Define a function

G( h ,k) := p Vg • h + div h — (p — 1)k 1 \ h \p 1

with k >  0 . We assume that there exist h o and k o > 0 such that

G (— o , k _o ) (x) ^ C 1 with some C 1 >  0 for all x E R N ,

(1.3)

(1.4)

(1.5)

(1.6)

G ( h o , k o') (x) от от as | x | от от ,

G (h o , k _o ) (x) >  C ₂ |V g(x) | ^p with some C 2 >  0 for all x E R N , there exist c 1 and R 1 such that

| D ^Y g(x) | C c 1 |V g(x) | | ^Y | for all | y | >  2, | x | >  R 1 .

Let g E C ^m ( R N , R + ) satisfy (1.3)-(1.6) and define

W^ ^p ( R N ) := { u E W m, ^p ( R N )} : i e ^pg^\ D k u \^P dx <  от ,

_R N    k =0

where Dk denote all distributional derivatives of order k, Wgm,p(RN)-norm is defined by lluHm,p,g,RN

1 m ^p / ^\ D k u \ ^p e ^pg dx )

_R N k =0

Denote W = W -’p (R N) П L g (R N).

Definition 1.1. We say that for a given u o E L g ( R N ) , u is an energy solution of (1.1), (1.2) for any T >  0 if

u E C([0, от; Lg (RN) П Lp(0,T; W))

and u satisfies (1.1) in sense of distributions and (1.2) holds.

The existence result of (1.1), (1.2) can be done exactly as in [3]. Note also that the following integration by parts formula occurs

t l/u™ epgdx - q/ho|q epg dx = -W^-u epg dxdT

(1.7)

RN              RN             0 RN

for all t >  0 . We say that a solution u satisfies the finite speed of propagation property (FSP for short) if from that for some fixed time t o u ( x, t o ) is compactly supported with respect to x follows that u(x,t) is compactly supported with respect to x for any t ^ t o . Let B r (0) be a ball of radius r centered at the origin. The main result of the paper reads as follows.

Theorem 1.1. Let u be an energy solution to (1.1), (1.2) in S t for any T >  0. Assume that 1 < q < p and support u g ( x ) C B _r 0 (0), r g <  от and (1.3)-(1.6) hold. Then u satisfies FSP and support of u ( x, t ) C B _r ( T ) (0) for any T >  0, and for T large enough

r ( T ) С C(ro, ||uo|| q,g) g ^(-1) (log T ).

(1.8)

Remark 1.1. The case m = 1 estimate (1.8) was obtained in [4]. As an example of g(s) we can choose g(s) = s ^a , 0 < s <  от , 1 С а < p/ ( p — 1) .

The problem of FSP has a long history. In practice, it arose during the first study of degenerate second-order parabolic equations. A large role in this was played by exact solutions of the self-similar type, giving an idea of the exact behavior of the radius of the support and the solution over time. Using the comparison method, estimates of the support for model equations were obtained. We refer the reader to the monograph [5] for a more detailed acquaintance with the history of the development of qualitative properties for various classes of nonlinear degenerate equations. In the early 80s, integral methods for estimating the support were proposed by Antontsev [6] and were further developed by Diaz and Veron [7]. For the first time, support radius estimates for high-order equations were obtained in the works of Bernis [8, 9].

Later, other energy approaches were proposed in the works of Shishkov [10, 11], Andreucci and the author [12–15]. A feature of the approach in these works is that a system of rings is considered as a parametrization of the domain. This allows one to study equations with nonpower weights and thus manages to bypass the Gagliardo–Nirenberg weighted inequalities.

In what we denote by C >  0 the constant which may vary from line to line and depends only on the data of the problem. The rest of the paper is organized as follows. In the Chapter 2 we give the proof of Theorem 1.1.

• 2.    Proof of Theorem 1.1

We start from the following Lemma [14].

Lemma 2.1. Let p >  0 and a 2 ^ a i + 1/2, 0 < а < a i be given. Define

S _p = {x G R N : (a i — a)p <  | x | < (a ₂ — a)p}.

Then there exists a Lipschitz-continuous function Z ^ 0 such that its support is S p , Z (x) = 1 in { a 1 p <  | x | < a ₂ p } , Z G C “ (S _p ), and \ D ^k Z | С Y(N k ) a ^— p ^{- k} in S _p . Moreover, for l >  m, 0 < k С m, 0 < E <  1, v G W ^m,p ( S _p ) , we have

( ap ) ^- П < l - k D m ^- v ^ ps p С E II C ' D m’U + YE -mk* a     Il C ' -m’U •     (2.D

Denote (Vg) := (1 + |Vg|p)1/p. It was proven in [1] that under the conditions (1.3)-(1.5) the following inequality holds j epg(Vg^mp |u|p dx С C

e ^pg | D ^m u | ^p dx.

RN

RN

Based on this inequality and condition (1.6), was shown that the following three properties are equivalent for any m ^ 2:

(i) u G w gm^,p ( R N ) ,

• (ii)    (V gp- k D k u G L p ( R N ) for 0 С k С m,

• ( iii )    (V g ) ^{m -} ‘ D ( e ⁹ u) G L p (R N) for 0 C l <  m. As a consequence the following norms

  D^ku \\^P Y , p,g

  
  

m

EIWm-k k=0

|ut'.*> := (ElIGg)”-Dk(egu)\p) ’ \ k=0                      / are equivalent to the norm W^'p (RN). Thus, using (2.1) with v = ue9, we get

(ap) ^-k \\ e ⁹ < ‘ ^-k D ^m-k u || ps p <  C \| e ⁹ ₍ ‘ D m u^ + Ct ^- m-k a e ₍ ‘ - m u^ .

(2.2)

For p >  4r) , define for 1/4 > P ₂ > P 1 >  0 , and i = 0,1,...

ρ r- = — i 2

^P 2 p + £ ^{( P} 2 - ^P 1 ⁾ ,  r i = P + ^P2P - p ^{( P} 2 - ^P ) .-

2 ⁱ                                        2 ⁱ

Define also the sequence of annuli A i = { r i <  | x | < ri' } C A i +1 . Let Z i , i ^ 1, be the function as in (2.2) , where P = P 2 — P 1 and

1P           P                      P a1 = 2 — P2 + 2i,  a = 2i+i,  a2 = 1 + P2— 2i ■

Then Zi = 1 in Ai, Zi =0 out of Ai+1, and \DkZi\ C C(2-ipP)-k. Next, choosing ZSu in the weak formulation of (1.1), (1.2) as a testing function, where s > m is large enough, we get

t sup ( |u|qZSep9dx + / [\Dmu\p Ziep9 dxdT

0^t C t J                 J J

R N             0 R N

C C ^ (2- i 0p ) k j у \ D ^m-k u \\ D ^m u \ ^{p -1} Z ^{s - k} e ^P9 dxdT. ^k =1           0 R N

Applying the Young inequality to the r. s. of (2.3), we have

t

• r. s. <  C^£ J j \D ^m u \^p Z S e p⁹ dxdT

0 RN

• + Ce^^ ^r ( 2 ^- i P p )~ ^Pp J J \ D ^{m - k} u \^P Z s ^{- kp} e ^p9 dxdT = J.

^{k =1}           0 R N

Setting s = Ip , l > m , and using (2.2), we bound above the second term in (2.4) we get the bound

t

J <  C p J J \ D^mu \^PCtg^pd dxdT

0 RN

+ Ce - p +M- m ( 2 ^-i Pp ) ^-mp J J | u | p C^^mp e p9 dxdT,

0 RN

(2.3)

(2.4)

(2.5)

where we choose e and E 1 so that Ce + Ce ^p+1 £ i < 1 . Thus, we arrive at

t sup [ |u|q Zipepg dx + / [ |Dmu|p Zlp epg dxdT

0 P t p t J                   J J

R ^N               0 R N

(2.6)

t

• <    C ( 2^{- i} Op^{) - mp} j У | u | ^p cf -^ e p⁹ dxdT.

0 R N

Let v G W ^m,p ( A ) is compactly support in A , then the Gagliardo-Nirenberg inequality yields

p

71vpdx,eifDmvpx+e- *C ; AA    A

Choosing v = (uZ i ^s ₊₁ )e ^g , we get with a = N ( p — q ) / ( N ( p - q) + mpq ) :

J | uZ i +i l p e ^pg dx

A i+1

P e¹ / l D ^m (( uZ S +1 ) e g )| ^p dx + e ^- 1 — a Cl  /

A i+1                                         A i+1

q e qg _d x

) q . (2.7)

Taking into account (i) - (iii) , we have

J | D m ( uZ^ ) e ^g | p dx P Cj | D ^m (( u4 +1 ))| ^p e ^9p dx.

A i+1                             A i+1

Thus, by (2.6), (2.7) we deduce

t

sup [\u(!\gepgdx + i /'|Dm(uZis) |pepg dxdT

0Pt ptJ                 J J

A i                   0 A i

t

P C (2 - i Op^{) - mp} I У | ( uZ _i ^s ₊₁ ⁾ | ^P e ^pg dxdT

⁰ A i+1

t

P e a C (2 ^{- i} 0p^{) - mp} J У ^|D ^{m (} uZ s ₊₁ ^)|P e pg dxdT

(2.8)

i

+ e-~ C (2-iOp)

0 Ai+i mp   0

A i+1

Choose e : e ^1/a C(2 i Op ) ^mp = 6. Then (2.8) reads:

t

Y i := sup У | uZ i |^q e^{p g} dx + У У | D ^m ( uZ is ) | ^P e^{p g} dxdT A i                       0 A i

p

P 6Y i +1 + ( 6 1-a C 2 i O ^-1) H p ^- H tl sup [\ u \e e q⁹ dx] ,

V 0 ^^TP t J          J

A∞ where H = N(p — q) + mpq. Hence, for small enough 5 by the standard iteration we get

p sup \\u\e epa dx < Cp-~ t\ sup [\u\q eq dx]

• 0 ^ t ^tj                         \ 0^t J             I

A 0                          A ∞

(2.9)

• <    Cd - ~ p - ~ te ^{-( p - q}' ^{) a (} 4 q( sup f \ u \ ^q e^{p g} dx] , ■ ^tj

A _∞

where A^ = {p(2 1 — #2) < \x\ < p(1 + #2)}, Ao = {p(2 1 — #1) < \x\ < p(1 + #1)}. Let

Then (2.9) gives

(2.10)

p > 4ro and p’ = p - u—, p«= p + u—,  n = 0,1,... n 2     2n     n         2n We choose d2 = a2-n,   d1 = a2-n-1,   и G (0, 4} . Define In := sup     (   \u\aepa dx. 0^t^t ρ′n<|x|<ρ′n′ nH H       Pn(P\P 1+ p-q

I n +1 <  C (u)2— p ^- -te r ^{(p - a}a (} 4q I n ^q .

Finally, by classical iterative Lemma 5.6 of [16, Chapter 2] one gets from (2.10) that In ^ 0 as n ^ ro, provided p-Hte-p-q"^4plO- < 51,                         (2.11)

where 5 1 is small enough and depends on the data of the problem only. Using (1.7), we have

Io <  l|uo||q,a = J \ u o \ ^q e ^{g ( x} dx <  ro .

R ^N

Thus, in order to check (2.11), it is enough to choose a free parameter ρ such that mpq           ρ                   q p p-q p Ne a(4)p t luola,a < 5p-q.

From this inequality for p >  1 and t > t o (l u o l q,a ) large enough, we conclude that the radius of support satisfies (1.8). Theorem 1.1 is proved.