Ukrainian Mathematical Journal, Vol. 58, No. 2, 2006
INITIAL-BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS WITH DAMPING. NEUMANN PROBLEM A. F. Tedeev
UDC 517.946
We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined.
1. Introduction In the present paper, we consider the Neumann problem N
ut =
∂
∑ ∂x (u m −1 Du λ −1ux ) − a( x ) Du ν
i =1
i
i
N
∑ u m −1 Du λ −1ux ni i
i =1
= 0,
u ( x, 0 ) = u0 ( x ),
q
,
( x, t ) ∈ Ω × 0, ( ∞ ),
( x, t ) ∈ ∂Ω × (0, ∞ ), x ∈ Ω,
(1.1)
(1.2)
(1.3)
where Ω ⊂ RN, N ≥ 2, is an unbounded domain with sufficiently smooth noncompact boundary and n = ( n i )
(
)
is the outer normal to ∂ Ω, Du = ux1 , … , ux N . In what follows, we assume that a ( x ) and u 0 ( x ) are nonnegative measurable functions and, in addition, u0 ∈ L1 ( Ω ), i.e., u0 has a finite mass. Moreover, m + λ – 2 ≥ 0,
λ > 0,
1 < q < λ + 1,
ν q > m + λ – 1.
(1.4)
We also assume that there are additional conditions imposed on the data of the problem. For the first time, the equation ut = Δu − Du
q
p
+ δu , δ > 0, was studied in [1] with an aim to analyze the influence of the term
− Du q (or, in other words, damping) on the problem of existence or nonexistence of global (in time) solutions of the Dirichlet problem. For the detailed analysis of the results accumulated in this field, see the survey [2]. We also mention a recent cycle of papers [3–6] and the bibliography therein. The aim of the present work is to determine conditions that should be imposed on q in order that the mass of the solution of problem (1.1)–(1.3) u(⋅, t ) 1, Ω ≡ u(⋅, t ) L1 (Ω) approach zero as t → ∞. Note that if a ( x ) ≡
u(⋅, t ) 1, Ω ≡ u0 1, Ω and, hence, the mass does not approach zero as t → ∞. However, it turns out that even the presence of a strong sink, i.e., damping, in (1.1) does not always guarantee 0, then, for almost all t > 0,
Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 272–282, February, 2006. Original article submitted October 10, 2005. 304
0041–5995/06/5802–0304
© 2006
Springer Science+Business Media, Inc.
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
WITH
DAMPING.
305
vanishing of the mass of the solution of problem (1.1)–(1.3). For a ( x ) ≡ const and Ω = RN (Cauchy problem), problem (1.1), (1.3) is studied in [7], where one can find an answer to the following question: What conditions should be imposed on the parameters of the problem, in order that the mass of the solution of the problem be vanishing? In fact, the authors of [7] managed to find the critical index q* = ( N ( m + λ − 1) + λ + 1) / ( N ν + 1) . This means that u(⋅, t ) 1, R N → 0 as t → ∞ for q ≤ q* and u(⋅, t ) 1, R N > c > 0 for q > q* and sufficiently large t > t0 . In addition, if, e.g., supp u0 < ∞, then the following estimates are proved for sufficiently large values of t : u(⋅, t )
1, R N
≤ c t –A
for
q < q* ,
(1.5)
where A =
q* − q ( N ν + 1) H
and
H = (λ + 1)( νq − 1) − q( m + λ − 2) ,
and u(⋅, t )
1, R N
≤ C(ln t )
−
1 νq −1
for
q = q* .
(1.6)
Note that problem (1.1)–(1.3) with a (x ) ≡ 0 was studied in [8–11], where the exact estimates were established for u(⋅, t ) ∞, Ω and the geometry of the support. Combining the approaches proposed in these papers and in [7], we establish estimates of the type (1.5) and (1.6) noticeably depending, as becomes clear in what follows, on the geometry of the domain and the behavior of a ( x ) at infinity. To formulate our principal results and prove them, we need several definitions and auxiliary statements. In what follows, by c we always denote constants depending only on parameters of the problem and independent of the size of the domain Qt . 2. Auxiliary Statements and Formulation of the Principal Results We select classes of domains with noncompact boundary satisfying the conditions of isoperimetric type. It is assumed that the origin of coordinates belongs to Ω. Let l ( v ) = inf { mes N −1 (∂Q ∩ Ω) } , where the infimum is taken over all open sets Q ⊂ Ω with Lipschitz boundary, and let mesN Q = v. We say that Ω belongs to the class B1 ( g ) if there exists a continuous function g ( v ) nondecreasing for all v > 0 such that
(
)1/ 2 ,
R ( v ) ≥ c0
v g(v)
not decrease for all v > 0. Further, let rk ( x ) = x12 +…+ xk2
v( N −1) / N does g(v)
1 ≤ k ≤ N, and let Ω ( ρ ) = Ω ∩
{ rk ( x ) < ρ }
for given ρ > 0 and V ( ρ ) = mesN Ω (ρ ). By R we denote the function inverse to V ( ρ ). We say that Ω belongs to the class B2 ( g ) if Ω ∈ B1 ( g ) and there exists a constant c0 > 0 such that
for all v > 0. It is easy to see that if Ω ∈ B1 ( g ), then the inequality
(2.1)
306
A. F. TEDEEV
R (v ) ≤ N
v , g(v)
(2.2)
opposite to inequality (2.1) is true for all v > 0. In addition, it follows from (2.1) and (2.2) that 1 1 ρg(V (ρ)) ≤ V ( ρ ) ≤ ρg(V (ρ)) N c0
(2.3)
for all ρ > 0. In turn, inequalities (2.3) imply that mesN Ω = ∞. The classes of domains B1 ( g ) and B2 ( g ) were introduced in [12], where the exact estimates were obtained for the rates of stabilization of solutions of the Neumann problem for linear second-order parabolic equations. A paraboloid-type domain Ωh = x ∈ R N : x ′ < x1h , where x ′ =
(
x22
+…+
)
1/ 2 x N2 ,
{
}
x1 > 1, 0 ≤ h ≤ 1, serves as a typical example of domains from the class B2 ( g ).
In this case (see [9] for N ≥ 2 and [12] for N = 2), we have
(
g ( v ) = c min v( N −1) / N , vγ
)
and
γ =
h( N − 1) . h( N − 1) + 1
(2.4)
Further, to avoid cumbersome formulations of the results, we assume that a ( x ) ≡ a ( rk ( x ) ). Moreover, let a ( ρ ) be an increasing function for all ρ > 0. If a1 ( s ) is a decreasing permutation of the function 1 / a ( rk ( x ) ), then, by definition, ⎡ 1 ⎤* 1 = a1 ( s ) = ⎢ . ⎥ ( a R (s)) ⎣ a(rk ( x )) ⎦
(2.5)
Recall that a decreasing permutation of a measurable function f ( x ) is defined as follows: f ( s )* = infτ { μ( τ) < s } , where μ ( τ ) = mesn { x ∈ Ω: f ( x ) > τ } (see, e.g., [13]). Assume that sq a(s)
increases for all
s > 0.
(2.6)
We now introduce the notion of solution of problem (1.1)–(1.3) in Q∞ = Ω × ( 0, ∞ ). We say that u ( x, t )
is a solution of problem (1.1)–(1.3) in Q ∞ if u ≥ 0, u ∈ L∞, loc (QT ) ∩ C((0, T ); L2, loc (Ω)) , | D uσ | λ + 1, a ( x ) | D uν | q ∈ L1, loc (QT ) , σ = ( m + λ – 1 ) / λ, and, for any T > 0 and any function η ∈ C01 (QT ),
∫∫ { − u ηt + u
QT
m −1
Du
λ −1
}
Du D η + a Du ν η d x d t = 0. q
(2.7)
In addition, u ( x, t ) → u0 as t → 0 in L1 ( Ω ). The problem of existence of solution of problem (1.1)–(1.3) is of independent interest. It will be investigated in a separate paper. Prior to give precise formulations of the principal results, we introduce necessary notation. Let P and ϕ
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
WITH
DAMPING.
307
be functions inverse to V ( s)m + λ − 2 s λ +1 and [s H a m + λ − 2 ( s)]1/( νq − ( m + λ −1)) , respectively, where, H = ( λ + 1 ) ( ν q – 1 ) – q ( m + λ – 2 ). The following theorems are the principal results of the present paper: Theorem 2.1. Assume that u ( x, t ) is a solution of problem (1.1) – (1.3) in Q∞ , suppu0 ⊂ Ω(ρ0 ) , ρ 0 < ∞, Ω ∈ B2 ( g ), and conditions (1.4) with m + λ – 2 > 0 and (2.6) are satisfied. Then the following estimates are true: E (t ) ≡
⎛ ϕ (t ) q ⎞ u ( ⋅ , t ) dx ≤ cV ϕ ( t ) ( ) ⎜ ⎟ ∫ ⎝ a(ϕ(t ))t ⎠ Ω
⎤ ⎡t a( P ( τ ) ) E (t ) ≤ c⎢ ∫ q dτ ⎥ νq −1 P( τ ) ⎥⎦ ⎢⎣ 1 V ( P( τ)) u( ⋅, t )
(
for all t > t0 = t0 ρ0 , u0
m+λ −2 1,Ω
∞, Ω
⎛ ϕ(t )λ +1 ⎞ ≤ c⎜ ⎟ t ⎝ ⎠
1 / ( νq −1)
,
(2.8)
−1 / ( νq −1)
,
(2.9)
1/ (m + λ − 2)
(2.10)
).
h Theorem 2.2. Assume that u ( x, t ) is a solution of problem (1.1) – (1.3) in Ω × ( 0, ∞ ), a ( x ) ≡ x1α , 0 ≤ α < q, suppu0 ⊂ Ω h (ρ0 ) and conditions (1.4) with m + λ – 2 > 0 and
q > q* =
Nh ( m + λ − 1) + λ + 1 + α , Nh ν + 1
(2.11)
(
where Nh = ( N – 1 ) h + 1, are satisfied. Then, for sufficiently large t > t1 = t1 ρ0 , u0
1,Ω
),
E ( t ) ≥ c > 0.
(2.12)
Theorem 2.3. Assume that u ( x, t ) is a solution of problem (1.1) – (1.3) in Q∞ , Ω ∈ B2 ( g ) and conditions (1.4) and (2.6) with ν = σ = ( m + λ – 1 ) / λ are satisfied. Then the following estimate is true: ⎤ ⎡ V (ρ(t )) E (t ) ≤ c⎢ u dx + ∫ 0 ρ(t )q / (q − λ ) a(ρ(t ))λ / (q − λ ) ⎥⎥ , ⎢⎣ Ω \ Ω(ρ(t )) ⎦
(2.13)
where ρ ( t ) is given by the formula sup p u0 ρ( valid for all t > 0.
q ( m + λ − 2 ) + ( λ +1)( q − λ )) / λ
a(ρ)
m+λ −2
= t ( q − λ )( m + λ −1) / λ
(2.14)
308
A. F. TEDEEV
We now present some examples. Let Ω = Ω h and let a ≡ x1α for x ≥ 1 and a ≡ 1 for 0 < x < 1 and 0 ≤ α < q. By using (2.8), we get E ( t ) ≤ c t – Λ,
(2.15)
(q* − q )( Nh ν + 1) , H1 = H + α ( m + λ – 2 ), and q* is specified by (2.11). It follows from (2.15) H1 that if q < q*, then E ( t ) → 0 as t → ∞. Under the same assumptions, for q = q*, inequality (2.9) yields the estimate where Λ =
E ( t ) ≤ c [ ln t ] − 1 / ( νq −1) .
(2.16)
Further, inequality (2.10) takes the form u( ⋅, t )
∞, Ω
≤ C t − ( λ +1+ α − q ) / H1 .
(2.17)
The results presented above show that q = q* plays the role of a critical index in problem (1.1)–(1.3). It is easy to see that this index depends not only on m , λ , and ν but also on the geometry of the domain and the function a ( x ). We also note that, for h = 1 and a ( x ) ≡ 1, the indicated new critical index coincides with the index obtained in [7] for the Cauchy problem. It is clear that the case of Cauchy problem is not excluded from our investigation. It should be emphasized that estimates (2.17) and (2.10) are independent of the geometry of the domain. The validity of the presented results is confirmed by the fact that the function u ( x, t ) = (t + t0 )− ( λ +1+ α − q ) / H1 f
( x (t + t0 )− ( νq − (m + λ −1)) / H ) 1
is a solution of Eq. (1.1) with a = x α . Moreover, f (r ) satisfies the equation d N −1 m −1 νq − ( m + λ − 1) ⎛ λ +1+ α − q ⎞ r f fr f (r ) + r fr ⎟ = r − ( N −1) −⎜ ⎝ ⎠ dr H1 H1
(
λ −1
)
fr – r α ( f ν )r . q
Further, Theorem 2.3 remains true in the nondegenerate case. Thus, if m + λ – 2 = 0, then ρ ( t ) = t1 / ( λ +1) , and if suppu0 < ∞, then it follows from (2.3) that
(
)
[(
E ( t ) ≤ cV t1 / ( λ +1) t − q / [ ( q − λ )( λ +1) ] a t1 / ( λ +1)
)]
− λ / (q − λ)
.
(2.18)
Note that, for ν = ( m + λ – 1 ) / λ, conditions (1.4) imply that λ < q < λ + 1. The critical role of the index q in inequality (2.18) is determined by the conditions of vanishing of the right-hand side of (2.18) as t → ∞. It seems likely that estimate (2.18) is new even for λ = 1. At the end of the section, we present the following auxiliary statement (special case of Lemma 3.1 in [14]): p Lemma 2.1. Assume that Ω ∈ B1 ( g ) and the following condition is satisfied: s a( s) increases for s > 0 and p > 1. Then the following inequality holds:
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
∫
a(rk ( x )) Du p dx ≥ c
Ω
Ep
(
G Eβp / ( p −β )
E βp / ( p −β )
WITH
DAMPING.
)
309
(2.19)
for p
1 ⎡ s ⎤ G (s ) = ⎢ , ⎥ ( g s a R ( ) ( s )) ⎣ ⎦
0 < β < p,
and
Eγ =
u γ dx .
∫
Ω
If sup pu( x ) ⊂ Ω(ρ) , then the Poincaré inequality
∫
u p dx ≤ c
Ω(ρ)
ρp a Du p d x a(ρ) Ω∫(ρ)
(2.20)
is an obvious corollary of inequality (2.19). 3. Proof of the Principal Results. Proof of Theorem 2.1 First, we prove that Z ( t ) = inf { r > 0 : u (x, t ) = 0 a.e. x ∈ Ω \ Ω ( r ) } ≤ c φ (t ).
(3.1)
Consider a sequence rn = 2 ρ ( 1 – 2– n – 1 ), n = 0, 1, … , ρ > 2ρ0 . Let ζ n (rk ( x )) be a sequence of smooth functions satisfying the conditions: ζ n = 0 for x ∈ Ω ( rn ), ζ n ≡ 1 for x ∈ Ω \ Ω(rn ) , where rn = 0.5 (rn + rn +1 ) , n –1 and | D ζn | ≤ c 2 ρ . Multiplying both sides of Eq. (1.1) by ζ λn +1 uθ , θ > 0, and integrating over Qt , we get
yn + 1 ≡ sup
1+ θ
∫u
0< τ
t
dx +
∫ ∫u
m +θ−2
Du
λ +1
d x dτ
0 Un t
+
∫ ∫
0 Un
uθ Du ν
q
d x dτ ≤ c
2 n( λ +1) ρλ +1
t
∫
∫
u m + λ + θ −1 d x dτ ,
(3.2)
0 Un \ Un
where Un = Ω \ Ω ( rn ) and Un = Ω \ Ω(rn ) . Just as in [9], we prove the inequality yn + 1 ≤ c
2 n( λ +1) (1+ θ)( λ +1) / K1+ θ 1+ ( m + λ − 2 )( λ +1) / K1+ θ t yn f0 t y0( m + λ − 2 ) / (1+ θ) , λ +1 ρ
(
where ( m + λ − 2 ) / (1+ θ ) 1 f0 ( s) = ⎡ F1( −1) ⎛⎝ ⎞⎠ s N (1+ θ) / K1+ θ ⎤ , ⎢⎣ ⎥⎦ s
K1 + θ = N ( m + λ – 2 ) + ( 1 + θ )( λ + 1 ),
)
(3.3)
310
A. F. TEDEEV
and F1( −1) is the function inverse to F1 ( s ) = s λ + ( λ +1) / β g( s − 1 )λ +1 . We rewrite inequality (3.3) in the form yna+1 ≤ c 2 n( λ +1) , A
(3.4)
where −1
⎛ ( m + λ − 2)(λ + 1)⎞ a = ⎜1 + ⎟ , ⎝ ⎠ K1+ θ A =
[t
(1+ θ )( λ +1) / K1+ θ
(
ρ− λ − 1 f0 t y0( m + λ − 2 ) / (1+ θ)
)]
−a
.
We need one more recurrence inequality. Since [9]
[ (
Z ( t ) ≤ c P t u0
m+λ −2 1,Ω
) + ρ0 ] ,
(3.5)
by using the Poincaré inequality (2.20) with p = q, we get t
∫ ∫
0 Un
t
w q d x dτ ≤ cρq ( a(ρ))− 1 ∫
∫
a Dw q d x dτ ≤ c
0 Un
Z (t ) q yn , a( Z ( t ) )
w = u( νq + θ) / q .
(3.6)
Hence, by applying the Hölder inequality to the right-hand side of (3.2) and using inequality (3.6), we obtain
yn + 1
2 n( λ +1) ≤ c λ +1 ρ
≤ c
t
∫ ∫u
m + λ + θ −1
d x dτ
0 Un
q 2 n( λ +1) ( νq − ( m + λ −1)) / ( νq + θ ) ⎡ Z (t ) ⎤ ρ tV ( ) [ ] ⎢ a( Z ( t ) ) ⎥ ρλ +1 ⎣ ⎦
( m + λ + θ −1) / ( νq + θ )
yn( m + λ + θ −1) / ( νq + θ) .
(3.7)
We now rewrite inequality (3.7) in the form ynb+1 ≤ c 2 n( λ +1) yn, B where b =
(3.8)
νq + θ > 1, because m + λ + θ −1 b
νq > m + λ – 1
and
⎧ Z (t ) q ⎫ B = ⎨ [ tV (ρ) ]( νq − ( m + λ −1)) / ( νq + θ) ρ− λ − 1 ⎬ . a( Z ( t ) ) ⎭ ⎩
Combining inequalities (3.4) and (3.8), in view of the Young inequality, we get
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
WITH
DAMPING.
311
ε1 ⎛ yna+1 ynb+1⎞ yn+ n( λ +1) 1 ≤ c + yn, ⎜ ⎟ ≤ c2 B ⎠ ⎝ A Aε1 B1− ε1
b < 1. Hence, by virtue of the iterative lemma 5.6 in [15, p. 113], we conclude that yn → 0, b +1− a
where ε1 = n → ∞, and
A( y0 B)(1− a ) / b ≤ c0 .
(3.9)
To estimate y0, we consider a sequence
y
(n)
1+ θ
∫
= sup
0 < τ < t r ( x )>ρ n k
u( ⋅, τ)
t
dx +
∫
∫
u m + θ − 2 Du
∫
uθ Du ν
λ +1
d x dτ
0 rk ( x ) > ρ n t
+
∫
q
d x dτ ,
ρn =
0 rk ( x ) > ρ n
Acting in exactly the same way as in the proof of (3.8), we obtain last inequality, we arrive at the estimate
y0 = y
(0 )
⎡ Z (t ) q ⎤ ≤ ctV (ρ) ⎢ ⎥ ⎣ a( Z (t )) ⎦
( y ( n ) )b ≤
( m + λ + θ −1) / ( νq − ( m + λ −1))
ρ(1 + 2 n ) . 2
c2 n( λ +1) y
( n+ 1 )
B. Integrating the
ρ− ( λ +1)( νq + θ) / ( νq − ( m + λ −1)) .
(3.10)
Substituting inequality (3.10) in (3.9) and taking into account the evident inequality Z ( t ) ≤ 2 ρ, we conclude H m+λ– 2 ≥ c1 t νq − ( m + λ −1) , which proves estimate (3.1). that u ≡ 0 for ρ a (ρ ) Further, we multiply both sides of Eq. (1.1) by uθ and integrate the result over Ω ( ρ ) with ρ = Z ( t ). This yields
1 d u1+ θ d x = – θ ∫ u m + θ − 2 Du θ + 1 dτ Ω(∫ρ) Ω(ρ) – c
∫
λ +1
dx
a Du( νq + θ) / q
q
dx ≤ – c
Ω(ρ)
∫
u
Ω(ρ)
( νq −1) / ( νq + θ )
d x ≤ V (ρ)
⎛ ⎞ νq + θ u d x ⎜⎜ ∫ ⎟⎟ ⎝ Ω( ρ) ⎠
a Du( νq + θ) / q
Ω(ρ)
By using the Hölder and Poincaré inequalities, we find
1+ θ
∫
(1+ θ ) / ( νq + θ )
q
dx.
(3.11)
312
A. F. TEDEEV
( νq −1) / ( νq + θ )
≤ cV (ρ)
⎛ − 1 (1+ θ ) / ( νq + θ )
[ ρ a(ρ) ] q
⎜⎜ ∫ a Du ⎝ Ω( ρ)
( νq + θ ) / q q
⎞ dx ⎟ ⎟ ⎠
(1+ θ ) / ( νq + θ )
.
Hence, it follows from (3.11) that ⎛ ⎞ d u1+ θ d x ≤ − cV (ρ)− ( νq −1) / (1+ θ) a(ρ)ρ− q ⎜ ∫ u1+ θ d x ⎟ ∫ dτ Ω(ρ) ⎝ Ω( ρ) ⎠
( νq + θ ) / (1+ θ )
.
Integrating this inequality, we easily get
∫
[
u1+ θ d x ≤ cV (ρ) ρq a(ρ)− 1 t −1
Ω(ρ)
]
(1+ θ ) / ( νq −1)
.
Finally, by using the Hölder inequality, in view of the last inequality, we obtain ⎛ ⎞ 1+ θ u ( ⋅ , t ) d x ≤ u d x ⎜ ⎟ ∫ ∫ ⎝ Ω( Z ( t ) ) ⎠ Ω( Z ( t ) ) ≤
[
1 / (1+ θ )
(
V ( Z (t ))θ / (1+ θ)
V ( Z (t )) Z (t )q a( Z (t ))−1 t −1
)(1+θ) / ( νq −1) ]
[
≤ cV (ϕ(t )) ϕ(t )q a(ϕ(t ))− 1 t −1
1 / (1+ θ )
]/
1 ( νq −1)
V ( Z (t ))θ / (1+ θ)
.
We prove estimate (2.9). Integrating Eq. (1.1) over Ω (ρ ), ρ = Z ( t ), we find d E (t ) = – D ( t ), dt
(3.12)
where E (t ) =
∫
u( ⋅, t ) d x
and
Ω(ρ)
D(t) =
∫
a Du ν
q
d x.
Ω(ρ)
It follows from the Hölder and Poincaré inequalities that ⎛ ⎞ E ( t ) ≤ ⎜ ∫ u νq d x ⎟ ⎝ Ω(ρ) ⎠
1 / ( νq )
( νq −1) / ( νq )
V (ρ)
1 / ( νq )
⎡ ρq ⎤ ≤ c⎢ D(t ) ⎥ ⎣ a(ρ) ⎦
V (ρ)( νq −1) / ( νq ) .
(3.13)
Since, for sufficiently large t > t0 , inequality (3.5) implies that Z ( t ) ≤ c P (t ), by using inequalities (3.12) and (3.13), we obtain
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
WITH
DAMPING.
d E (t ) a( P ( t ) ) ⎡ ⎤ νq ≤ –c ⎢ q ⎥ E (t ) . ν q − 1 dt P( t ) ⎦ ⎣ V ( P ( t ))
313
(3.14)
Hence, integrating inequality (3.14) from t0 to t, we get ⎡t ⎤ a( P(τ)) ⎥ τ E (t ) ≤ c ⎢ ∫ d q ⎢⎣ t V ( P(τ)) νq −1 P(τ) ⎥⎦ 0
−1 / ( νq −1)
.
Theorem 2.1 is proved. Proof of Theorem 2.2. For any 0 < t1 < t, we have t
E ( t1 ) = E ( t ) +
α
Du ν
∫ ∫ x1
q
d x dτ.
(3.15)
t1 Ω
By using the Hölder inequality, we get t
∫∫
t1 Ω
x1α
Du
ν q
⎡t d x dτ ≤ ⎢ ∫ ⎢⎣ t 1
∫u
m +θ−2
Du
λ +1
Ω
⎡t × ⎢∫ ⎢⎣ t 1
⎤ d x dτ ⎥ ⎥⎦
q / ( λ +1)
⎤ α ( λ +1) / ( λ +1− q ) q (( λ +1) ν − ( m + λ −1) − θ ) / ( λ +1− q ) ⎥ x u d x d τ ∫ 1 ⎥⎦ Ω
( λ +1− q ) / ( λ +1)
≡ J1q / ( λ +1) J2( λ +1− q ) / ( λ +1) , (3.16) where θ > 0 is a sufficiently small number. Multiplying both sides of Eq. (1.1) by uθ and integrating over Ω × ( t1 , t ), we find J1 ≤ c ∫ u1+ θ ( ⋅, t1 ) d x .
(3.17)
Ω
In what follows, we need the following estimate for Ω ∈ B1 [8, 9]:
u( ⋅, t )
∞, Ω
≤ c
t −1 ∫
t
t /2
E ( τ ) dτ
m+λ −2 ⎞ t ⎛ ψ ⎜ t ⎛⎝ t − 1 ∫ E( τ) dτ⎞⎠ ⎟ t /2 ⎝ ⎠
where ψ is the function inverse to Ψ ( z ) = z m+ λ − 2 ( z / g( z ))λ +1.
,
(3.18)
In particular, if Ω = Ωh, then, by virtue of (2.4), we have ψ ( z ) = cz (( m + λ − 2 ) Nh + λ +1) / Nh for z > 1. Hence,
314
A. F. TEDEEV
in view of the fact that E (t ) ≤ u0
1,Ω
∀ t > 0, it follows from inequality (3.18) for t > 1 that
u( ⋅, t )
∞, Ω
λ +1) / K h − N h / K h ≤ c u0 1( ,Ω t ,
(3.19)
where Kh = ( m + λ – 2 ) Nh + λ + 1. Thus, we have 1+ θ
∫u
( ⋅, t ) d x ≤ E(t1 ) u( ⋅, t1 )
Ω
θ ∞, Ω
≤ cE(t1 ) u0 1( ,λΩ+1)θ / K h t − Nh θ / K h .
(3.20)
We also note that
(
Z (t ) ≤ c ρ0 + u0 h for Ω = Ω and t > t0 = ρ0K h u0
( m + λ − 2) / Kh 1 / Kh t 1, Ω
m+λ −2 . 1,Ω
)
≤ 2c u0
( m + λ − 2) / Kh 1 / Kh t 1, Ω
≡ Z˜ (t )
Therefore, for t0 < t1 < t, we can write
t
J2 ≤ c ∫ E( τ) Z˜ ( τ)α ( λ +1) / ( λ +1− q ) u( ⋅, τ)
( H − θq ) / ( λ +1− q ) dτ ∞, Ω
t1
t
m + λ − 2 )α + ( H − θq )( λ +1)) / ( K h ( λ +1− q )) ≤ c u0 1((,Ω E(t1 ) ∫ τ −( HNh − θqNh − α ( λ +1)) / [( λ +1− q ) K h ] dτ.
(3.21)
t1
Since q > q* , the integral on the right-hand side of (3.21) converges for θ <
(q − q* )( Nh ν + 1) . CombinqNh
ing estimates (3.15)–(3.17) and (3.21), we obtain E ( t1 ) ≤ E ( t ) + c* u0
H / Kh − ( q − q* ) Nh / Kh E(t1 ) 1, Ω t1
(3.22)
for all t1 > t0 . We now choose t1 as follows:
(
⎧ t1 = max ⎨ t0 , 2c* u0 ⎩
)
[
* H1 / K h K h / ( q − q )( N h ν +1) 1, Ω
] ⎫. ⎬ ⎭
As a result, inequality (3.22) readily implies that 2 E ( t ) ≥ E ( t1 ) for all t > t1 . We also note [7] (Lemma 4.1) that the case where u ( x, t1 ) ≡ 0 for all x ∈ RN and some t1 > 0 does not take place. Theorem 2.2 is proved. Proof of Theorem 2.3. We have E (t ) =
∫ u( ⋅, t ) d x
Ω
=
∫
u( ⋅, t ) d x +
Ω(ρ)
∫
u( ⋅, t ) d x ≡ I1 + I2 .
Ω \ Ω( ρ)
(3.23)
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
WITH
DAMPING.
315
According to the Hölder inequality, we can write ⎛ ⎞ I1 ≤ ⎜ ∫ u ν q d x ⎟ ⎝ Ω(ρ) ⎠
1 / ( νq )
V (ρ)( νq −1) / ( νq ) .
(3.24)
Let uν = v. Then, by using Lemma 2.1 with β = ν – 1 < q, p = q, we get
∫ a Dv
D (t) ≡
q
dx ≥ c
Ω
G
(
Fq (t ) νq / ( νq −1) E (t ) Fq (t )− 1 / ( νq −1)
)
,
(3.25)
where Fq ( t ) =
∫ v(⋅, t )
q
q
dx
⎡ s ⎤ ( a( R( s)))−1 . G(s ) = ⎢ ⎣ g( s) ⎥⎦
and
Ω
Recall that
s ∼ R ( s ) for Ω ∈ B2 ( g ). It follows from inequality (3.25) that g( s )
Fq ( t ) ≤ c E ( t )
where G1 ( s ) = obtain
νq ⎧
( −1) ⎛ ⎨ G1 ⎜
⎩
D(t ) ⎞ ⎫ ⎟⎬ ⎝ E (t ) ν q ⎠ ⎭
νq −1
,
(3.26)
s νq −1 ( −1) is the function inverse to G1 . Further, integrating Eq. (1.1) over Ω, we −1 and G1 G( s )
d E(t ) = – D ( t ). Therefore, relations (3.23), (3.24), and (3.26) imply the inequality dt
{
d E ( τ ) ≤ c E( τ) V (ρ)( νq −1) / ( νq ) G1( −1) ⎛ ⎛ − E( τ) ⎞ E( τ)− ( νq ) ⎞ ⎝ ⎝ dτ ⎠ ⎠
}
( νq −1) / ( νq )
+ I2 ( τ ).
(3.27)
To estimate I2 ( τ ), we act as follows: Let ζ ( x ) = ζ ( rk ( x )) be a smooth function equal to 1 for rk ( x ) > ρ, to 0 for rk ( x ) < ρ / 2, and such that 0 ≤ ζ ≤ 1 for 0 < rk < ∞. Moreover, | D ζ | ≤ c / ρ. We now multiply both sides of Eq. (1.1) by ζ s, s ≥ λ + 1 and integrate the result over Ω. This yields d s ∫ ζ u(⋅, τ) d x + dτ Ω
∫a
Du ν ζ s d x ≤ q
Ω
ν =
c ρ Ω( ρ)
∫
ζ s −1 Du ν
\ Ω(ρ \ 2 )
m + λ −1 . λ
By applying the Young inequality to the right-hand side of this inequality, we get
q
dx,
316
A. F. TEDEEV
d s ∫ ζ u( x , τ ) d x + dτ Ω
∫a
Du ν ζ s d x q
Ω
λ ≤ c ε q / λ ∫ a Du ν q Ω
q
dx + c
q − λ ε −( q − λ ) / λ q ρq / ( q − λ )
∫
a(rk ( x )) − λ / ( q − λ ) d x .
Ω( ρ) \ Ω(ρ \ 2 )
λ We now set c ε q / λ = 1 / 2. As a result, the last inequality implies that q d V (ρ) . ζ su( ⋅, τ) d x ≤ c q / ( q − λ ) ∫ dτ Ω ρ a(ρ)λ / ( q − λ ) Integrating this inequality, we get
∫ ζ u( x , τ ) d x s
Ω
≤
∫ ζ u0 d x s
+ c
Ω
ρ
τV (ρ) ≡ F˜ (ρ, τ) ≤ F˜ (ρ, t ) , a(ρ)λ / ( q − λ )
q / (q − λ)
t1 < τ < t.
(3.28)
Further, we note that
[
]
G1 ( s ) = s ε 0 s( N −1) / N g( s − 1 ) ,
Therefore,
q
where
N ( νq − 1) + q . N
ε0 =
G1 ( s) increases. This means that the function sε 0
(
s G1( − 1) ( s − νq )
)( νq −1) / ( νq)
= s qN / ε 0
[ (G
)
( − 1) − νq ( νq −1) / ( νq ) ( νq −1) / ε 0 (s ) s 1
]
also increases. It follows from (3.27) and (3.28) that νq ( νq −1) dy ⎛ y( τ ) ⎞ / ⎞ V (ρ)− 1⎟ ≤ − c E(t1 ) νq − νqε 0 / ( νq −1) y( τ) νqε 0 / ( νq −1) G1(V (ρ)− 1 ) , ≤ − c E(t1 ) νq G1 ⎜ ⎛ ⎝ ⎠ ⎝ E(t1 ) ⎠ dτ
(3.29)
where y (τ ) = E ( τ ) – F˜ (ρ, t ) . Hence, for t1 = σ t, σ ∈ (0, 1), we integrate inequality (3.29) from σ t to t and obtain ( νq −1) / ( νqε 0 − νq +1)
E ( t ) ≤ c E(σt )δ σ − ( νq −1) / ( νqε 0 − νq +1)t − ( νq −1) / ( νqε 0 − νq +1) G1(V (ρ)− 1 ) where δ =
+ F˜ (ρ, t ) ,
νq 2 < 1. Finally, integrating inequality (3.30) with respect to σ, we find νq 2 + N ( νq − 1)2 E ( t ) ≤ c t − 1 / ( νq −1) G1(V (ρ)− 1 )
− 1 / ( νq −1)
+ c F˜ (ρ, t ) .
(3.30)
INITIAL-BOUNDARY-V ALUE PROBLEMS FOR QUASILINEAR DEGENERATE HYPERBOLIC EQUATIONS
WITH
DAMPING.
317
We now note that G1(V (ρ)− 1 ) ∼ V (ρ) ρq / ( νq −1) a(ρ)− 1 / ( νq −1) and determine ρ from the condition ρ(t )( q( m + λ − 2 ) + ( λ +1)( q − λ )) / λ a(ρ(t ))m + λ − 2 = t ( q − λ )( m + λ −1) / λ . This yields the required statement. Theorem 2.3 is proved. The present work was supported by the INTAS (Grant No. 03-51-5007). REFERENCES 1. M. Chipot and F. B. Weissler, “Some blow-up results for a nonlinear parabolic equation with a gradient term,” SIAM J. Math. Anal., 20, 886–907 (1989). 2. P. Souplet, “Recent results and open problems on parabolic equations with gradient nonlinearities,” J. Different. Equat., No. 20, 1–19 (2001). 3. P. Laurencot and P. Souplet, On the Growth of Mass for a Viscous Hamiltonian–Jacobi Equation, Preprint (2002). 4. M. Ben-Arti, P. Souplet, and F. B. Weissler, “The local theory for viscous Hamiltonian–Jacobi equations in Lebesgue spaces,” J. Math. Pures Appl., 81, 343–378 (2002). 5. P. Souplet, Gradient Blow-Up for Multidimensional Nonlinear Parabolic Equations with General Boundary Conditions, Preprint (2002). 6. S. Benachour, P. Laurencot, D. Schmidt, and P. Souplet, Extinction and Nonextinction for Viscous Hamilton–Jacobi Equations in R N, Preprint (2002). 7. D. Andreucci, A. F. Tedeev, and M. Ughi, “The Cauchy problem for degenerate parabolic equations with source and damping,” Ukr. Math. Bull., 1, No. 1, 1–23 (2004). 8. A. F. Tedeev, “Estimates of the rate of stabilization of the solution of the second mixed problem for a second-order quasilinear parabolic equation as t → ∞,” Differents. Uravn., 27, No. 10, 1795–1806 (1991). 9. D. Andreucci and A. F. Tedeev, “A Fujita-type results for degenerate problem in domains with noncompact boundary,” J. Math. Anal. Appl., 231, 543–567 (1999). 10. D. Andreucci and A. F. Tedeev, “Optimal bounds and blow-up phenomena for parabolic problems in narrowing domains,” Proc. Roy. Soc. Edinburgh A, 128, No. 6, 1163–1180 (1998). 11. D. Andreucci and A. F. Tedeev, “Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity,” Adv. Different. Equat., 5, 833–860 (2000). 12. A. K. Gushchin, “On the estimates of solutions of boundary-value problems for second-order parabolic equations,” Tr. Mat. Inst. Akad. Nauk SSSR, 126, 5–45 (1973). 13. G. Talenti, “Elliptic equations and rearrangements,” Ann. Scuola Norm. Super. Pisa, 4, No. 3, 697–718 (1976). 14. D. Andreucci, G. R. Cirmi, S. Leonardi, and A. F. Tedeev, “Large time behavior of solutions to the Neumann problem for quasilinear second-order degenerate parabolic equations in domains with noncompact boundary,” J. Different. Equat., 174, 253–288 (2001). 15. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1967).
