Abstract
In this article, we study the quasilinear parabolic problem
where Ω is a bounded domain in ℝ^{N}, m > 0 and g(u) satisfies g(u) ≤ K_{1}u^{1+ν }with 0 ≤ ν < m. By the Moser's technique, we prove that if α, β > 1, 0 ≤ p < q, 1 ≤ q < m + 2, p + α < q + β, there exists a weak solution for all u_{0 }∈ L^{1}(Ω). Furthermore, if 2q ≤ m + 2, we derive the L^{∞ }estimate for ∇u(t). The asymptotic behavior of global weak solution u(t) for small initial data u_{0 }∈ L^{2}(Ω) also be established if p + α > max{m + 2, q + β}.
2000 Mathematics Subject Classification: 35K20; 35K59; 35K65.
Keywords:
quasilinear parabolic equation; L^{∞ }estimates; asymptotic behavior of solution1 Introduction
In this article, we are concerned with the initial boundary value problem of the quasilinear parabolic equation with nonlinear gradient term
where Ω is a bounded domain in ℝ^{N }with smooth boundary ∂Ω and m > 0, α, β > 1, 0 ≤ p < q, 1 ≤ q < m + 2.
Recently, Andreu et al. in [1] considered the following quasilinear parabolic problem
where α, β > 1, 0 ≤ p < q ≤ 2, p + α < q + β and u_{0 }∈ L^{1}(Ω). By the socalled stability theorem with the initial data, they proved that there exists a generalized solution u(t) ∈ C([0, T], L^{1}) for (1.2), in which u(t) satisfies and
for ∀t ∈ [0, T] and , where Q_{T }= Ω × (0, T], and for any k > 0,
J_{k}(u) is the primitive of A_{k}(u) such that J_{k}(0) = 0. The problem similar to (1.2) has also been extensively considered, see [26] and the references therein. It is an interesting problem to prove the existence of global solution u(t) of (1.2) or (1.1) and to derive the L^{∞ }estimate for u(t) and ∇u(t).
Porzio in [7] also investigated the solution of LerayLions type problem
where a(x, t, s, ξ ) is a Carathéodory function satisfying the following structure condition
with θ > 0 and u_{0 }∈ L^{q}(Ω), q ≥ 1. By the integral inequalities method, Porzio derived the L^{∞ }decay estimate of the form
with C = C(N, q, m, θ), α = mq(N(m  2) + mq)^{1}, λ = N(N(m  2) + mq)^{1}.
In this article, we will consider the global existence of solution u(t) of (1.1) with u_{0 }∈ L^{1}(Ω) and give the L^{∞ }estimates for u(t) under the similar condition in [1]. More specially, we will study the behavior of solution u(t) as t → 0^{+}. Obviously, if m = 0 and g ≡ 0, problem (1.1) is reduced to (1.2). We remark that the methods used in our article are different from that of [1]. In L^{∞ }estimates, we use an improved Morser's technique as in [810]. Since the equation in (1.1) contains the nonlinear gradient term uu^{α2}∇u^{p }and uu^{β2}∇u^{q}, it is difficult to derive L^{∞ }estimates for u(t) and ∇u(t).
This article is organized as follows. In Section 2, we state the main results and present some Lemmas which will be used later. In Section 3, we use these Lemmas to derive L^{∞ }estimates of u(t). Also the proof of the main results will be given in Section 3. The L^{∞ }estimates of ∇u(t) are considered in Section 4. The asymptotic behavior of solution for the small initial data u_{0}(x) is investigated in Section 5.
2 Preliminaries and main results
Let Ω be a bounded domain in ℝ^{N }with smooth boundary ∂Ω and ∥·∥_{r}, ∥·∥_{1,r }denote the Sobolev space L^{r}(Ω) and W^{1,r}(Ω) norms, respectively, 1 ≤ r ≤ ∞. We often drop the letter Ω in these notations.
Let us state our precise assumptions on the parameters p, q, α, β and the function g(u).
(H_{1}) the parameters α, β > 1, 0 ≤ p < q < m + 2 < N, p + α < q + β and q(α  1) ≥ p(β  1),
(H_{2}) the function g(u) ∈ C^{1 }and ∃K_{1 }≥ 0 and 0 ≤ ν < max{q + β  2, m], such that
(H_{3}) the initial data u_{0 }∈ L^{1}(Ω),
(H_{4}) 2q ≤ 2 + m, α, β < 2 + m(1 + 1/N)/2,
(H_{5}) the mean curvature of H(x) of ∂Ω at x is nonpositive with respect to the outward normal.
Remark 2.1 The assumptions (H_{1}) and (H_{3}) are similar to as in [1].
Definition 2.2 A measurable function u(t) = u(x, t) on Ω × [0, ∞) is said to be a global weak solution of the problem (1.1) if u(t) is in the class
and , and for any the equality
is valid for any T > 0.
Remark 2.3 In [1], the concept of generalized solution for (1.2) was introduced. A similar concept can be found in [7,11]. By the definition, we know that weak solution is the generalized solution. Conversely, a generalized solution is not necessarily weak solution.
Our main results read as follows.
Theorem 2.4 Assume (H_{1})(H_{3}). Then the problem (1.1) admits a global weak solution u(t) which satisfies
and the estimates
Furthermore, if (H_{4}) is satisfied, the solution u(t) has the following estimates
with r > λ = N(mN + m + 2)^{1 }and C_{0 }= C_{0}(T, ∥u_{0}∥_{1}).
Theorem 2.5 Assume (H_{1})(H_{5}). Then the solution u(t) of (1.1) has the following L^{∞ }gradient estimate
with σ = (2 + 2λ + N)(mN + 2m + 4)^{1 }and C_{0 }= C_{0}(T, ∥u_{0}∥_{1}).
Remark 2.6 The estimates (2.3) and (2.6) give the behavior of ∥u(t)∥_{∞ }and ∥∇u(t)∥_{∞ }as
Theorem 2.7 Assume the parameters α, β > 1, γ ≥ 0, 0 ≤ q < m + 2 < N and p < m + 2 < p + α, α ≤ (m + 2  p)(1 + 2N^{1}).
Then, ∃d_{0 }> 0, such that u_{0 }∈ L^{2}(Ω) with ∥u_{0}∥_{2 }< d_{0}, the initial boundary value problem
admits a solution , which satisfies
where C = C(∥u_{0}∥_{2}).
Theorem 2.8 Assume the parameters γ > 0, α, β > 1, 1 ≤ p < q < m + 2 < N and τ = N(μ  q)(q + β) ≤ 2(q^{2 }+ Nβ) with μ = (qα  pβ)/(q  p) > q + β.
Then, ∃d_{0 }> 0, such that u_{0 }∈ L^{2 }with ∥u_{0}∥_{2 }< d_{0}, the initial boundary value problem
admits a solution which satisfies
where C = C(∥u_{0}∥_{2}).
To obtain the above results, we will need the following Lemmas.
Lemma 2.9 (GagliardoNirenberg type inequality) Let β ≥ 0, N > p ≥ 1, q ≥ 1 + β and 1 ≤ r ≤ q ≤ pN(1 + β)/(N  p). Then for u^{β}u ∈ W^{1,p}(Ω), we have
with θ = (1 + β)(r^{1 } q^{1})/(N^{1 } p^{1 }+ (1 + β)r^{1}), where the constant C_{0 }depends only on p, N.
The Proof of Lemma 2.9 can be obtained from the wellknown GagliardoNirenbergSobolev inequality and the interpolation inequality and is omitted here.
Lemma 2.10 [10] Let y(t) be a nonnegative differentiable function on (0, T] satisfying
with A, θ > 0, λθ ≥ 1, B, C ≥ 0, k ≤ 1. Then, we have
3 L^{∞ }estimate for u(t)
In this section, we derive a priori estimates of the assumed solutions u(t) and give a proof of Theorem 2.4. The solutions are in fact given as limits of smooth solutions of appropriate approximate equations and we may assume for our estimates that the solutions under consideration are sufficiently smooth.
Let and u_{0,i }→ u_{0 }in L^{1}(Ω) as i → ∞. For i = 1, 2, ..., we consider the approximate problem of (1.1)
The problem (3.1) is a standard quasilinear parabolic equation and admits a unique smooth solution u_{i}(t)(see Chapter 6 in [12]). We will derive estimates for u_{i}(t). For the simplicity of notation, we write u instead of u_{i }and u^{k }for u^{k1}u where k > 0. Also, let C, C_{j }be generic constants independent of k, i, n changeable from line to line.
Lemma 3.1 Let (H_{1})(H_{3}) hold. Suppose that u(t) is the solution of (3.1), then u(t) ∈ L^{∞}([0, ∞), L^{1}).
Proof Let n = 1, 2, ..., and
It is obvious that f_{n}(s) is odd and continuously differentiable in ℝ^{1}. Furthermore, and f_{n}(s) → sign(s) uniformly in ℝ^{1}.
Multiplying the equation in (3.1) by f_{n}(u) and integrating on Ω, we get
and the application of the Young inequality gives
where μ = (qα  pβ)(q  p)^{1 }≥ 1, i.e q(α  1) ≥ p(β  1).
In order to get the estimate for the third term of lefthand side in (3.2), we denote
It is easy to verify that F_{n}(u) is odd in ℝ^{1}. Then, we obtain from the Sobolev inequality that
with some λ_{0 }> 0 and
We note that F_{n}(u)^{q }≤ n^{(q+β1) }in and
On the other hand, we have u(x, t) ≥ n^{1 }in Ω_{n }and
This implies that there exists λ_{1 }> 0, such that
Then it follows from (3.4)(3.5) that
with some C_{2 }> 0.
Similarly, we have from the assumption (H_{2}) and the Young inequality that
Furthermore, the assumption μ < q + β implies that
Then (3.2)(3.3) and (3.6)(3.8) give that
Letting n → ∞ in (3.9) yields
Note that
with some λ_{2 }> 0. Then (3.10) becomes
This gives that u(t) ∈ L^{∞}([0, ∞), L^{1}) if u_{0 }∈ L^{1}.
Remark 3.2 The differential inequality (3.10) implies that the solution u_{i}(t) of (3.1) satisfies
withC_{0 }= C_{0}(T, ∥u_{0}∥_{1}).
Lemma 3.3 Assume (H_{1})(H_{4}). Then, for any T > 0, the solution u(t) of (3.1) also satisfies the following estimates:
where λ = N(mN + m + 2)^{1}, C_{0 }= C_{0}(T, ∥u_{0}∥_{1}).
Proof Multiplying the equation in (3.1) by u^{k1}, k ≥ 2, we have
It follows from the Hölder and Sobolev inequalities that
in which θ_{1 }= kλ(m  ν + (m + 2)N^{1}), θ_{2 }= νλ(m + 2)N^{1}, θ_{3 }= νλ(k + m), σ = νλ, s = N(k + m)(N  m  2)^{1}.
Note that
and
with some C_{1 }independent of k and μ = (qα  pβ)(q  p)^{1 }< q + β.
Without loss of generality, we assume k > 3  μ. Similarly, we derive
with ξ_{1 }= sup_{t≥0}∥u(t)∥_{1 }and
Then, for any η > 0,
with μλ_{0}θ = 1, (1  μλ_{0})θ' = 1.
Note that μ_{1}θ' < k. Let . Then it follows from (3.15) that
with γ = qθ'θ^{1 }= qμλ_{0}/(1  μλ_{0}). Then, (3.14) becomes
or
with σ_{0 }= max{σ, γ} = max{νλ, γ}.
Now we employ an improved Moser's technique as in [8,9]. Let {k_{n}} be a sequence defined by k_{1 }= 1, k_{n }= R^{n2}(R  m  1) + m(R  1)^{1}(n = 2, 3, ...) with R > max{m + 1, m + 4  μ} such that k_{n }≥ 3  μ(n ≥ 2). Obviously, k_{n }→ ∞ as n → ∞.
By Lemma 2.9, we have
Then, inserting (3.18) into (3.17) (k = k_{n}), we find that
or
where . It is easy to see that
Denote
Then (3.20) can be rewritten as follows
We claim that there exist a bounded sequence {ξ_{n}} and a convergent sequence {λ_{n}}, such that
Indeed, by Lemma 3.1, the estimate (3.22) holds for n = 1 if we take λ_{1 }= 0, ξ_{1 }= sup_{t≥0 }∥u(t)∥_{1}. If (3.22) is true for n  1, then we have from (3.21) and (3.22) that
where
Applying Lemma 2.10 to (3.23), we have
This implies that for t ∈ (0, T),
where
in which the fact k_{n }~ β_{n }as n → ∞ has been used.
It is not difficult to show that {ξ_{n}} is bounded. Furthermore, by Lemma 4 in [9], we have
Letting n → ∞ in (3.22) implies that (3.13) and we finish the Proof of Lemma 3.3.
Lemma 3.4. Let (H_{1})(H_{4}) hold. Then, the solution u(t) of (3.1) has the following estimates
and
with r > λ = N(mN + m + 2)^{1}, C_{0 }= C_{0}(T, ∥u_{0}∥_{1}).
Proof We first choose r > λ and η(t) ∈ C[0, ∞) ⋂ C^{1}(0, ∞) such that η(t) = t^{r }when t ∈ [0, 1]; η(t) = 2, when t ≥ 2 and η(t), η'(t) ≥ 0 in [0, ∞). Multiplying the equation in (3.1) by η(t)u, we have
Note that
Hence, we have
By Lemma 3.1 and the estimate (3.13), we get
Since μ < q + β, we have from Sobolev inequality that
Similarly, we have from 2 + ν < q + β that
Therefore, it follows from (3.30)(3.33) that
Next, let . Furthermore, multiplying the equation in (3.1) by ρ(t)u_{t }yields
By the assumption p < q and the Cauchy inequality, we deduce
and
and
with h(t) = ∥u(t)∥_{∞}.
Now, it follows from (H_{4}) and (3.35)(3.38) that
or
where C_{0 }= C_{0}(T, ∥u_{0}∥_{1}) and the fact 2 + λ ≥ 2(μ  1)λ has been used.
Since the function h^{2(β1)}(t) ∈ L^{1}([0, T]), the application of the Gronwall inequality to (3.40) gives
Hence,
and the Proof of Lemma 3.4 is completed.
Proof of Theorem 2.4 Noticing that the estimate constant C_{0 }in (3.12)(3.13) and (3.27)(3.28) is independent of i, we have from the standard compact argument as in [1,13,14] that there exists a subsequence (still denoted by u_{i}) and a function satisfying
Since is bounded in , we see further that
for some . As the Proof of Theorem 1 in [9], we have χ = A(u) = div((∥∇u∥^{m}∇u).
Then, the function u is a global weak solution of (1.1). Furthermore, it follows from Lemma 3.4 that u(t) satisfies the estimate (2.4)(2.5). The Proof of Theorem 2.4 is now completed.
4 L^{∞ }estimate for ∇u(t)
In this section, we use an argument similar to that in [9,10,15] and give the Proof of Theorem 2.5. Hence, we only consider the estimate of ∥∇u∥_{∞ }for the smooth solution u(t) of (3.1). As above, let C, C_{j }be the generic constants independent of k and i. Denote
Multiplying (3.1) by div(∇u^{k2}∇u), k ≥ m + 2 and integrating by parts, we have
Since
we have
and
Similarly, we obtain the following estimates
and
where h(t) = ∥u(t)∥_{∞ }≤ Ct^{λ}.
Moreover, we assume that 2q ≤ m + 2, 2p ≤ m + 2, then (4.1) becomes
where h_{1}(t) = max{h^{2(α1)}(t), h^{2(β1)}(t), h^{ν}(t)}. Since , we get h_{1}(t) ∈ L^{1}([0.T]) for any T > 0.
If H(x) ≤ 0 on ∂Ω and N > 1, then by an argument of elliptic eigenvalue problem in [15], there exists λ_{1 }> 0, such that
Hence, by (4.7) and (4.8), we see that there exists C_{1 }and C_{2 }such that
Let k_{1 }= m + 2, R > m + 1, k_{n }= R^{n2 }(R1m) + m (R1)^{1}, , n = 2, 3,.... Then, the application of Lemma 2.9 gives
Inserting this into (4.9)(k = k_{n}), we get
By (3.28), we take y_{1 }= max{1, C_{0}}, z_{1 }= (1 + λ)/(m + 2). As the Proof of Lemma 3.3, we can show that there exist bounded sequences y_{n }and z_{n }such that
in which z_{n }→ σ = (2 + 2λ + N)(mN + 2m + 4)^{1}. Letting n → ∞ in (4.12), we have the estimate (2.6). This completes the Proof of Theorem 2.5.
5 Asymptotic behavior of solution
In this section, we will prove that the problem (1.1) admits a global solution if the initial data u_{0}(x) is small under the assumptions of Theorems 2.7 and 2.8. Also, we derive the asymptotic behavior of solution u(t).
Proof of Theorem 2.7 The existence of solution for (1.1) in small u_{0 }can be obtained by a similar argument as the Proof of Theorem 2.4. So, it is sufficient to derive the estimate (2.8).
Multiplying the equation in (2.7) by u and integrating over Ω, we obtain
Since p < m + 2 < p + α, it follows from Lemma 2.9 that
with
The assumption on α shows that r ≤ 2. Then, (5.1) can be rewritten as
By the Sobolev embedding theorem,
we obtain from (5.3) and (5.4) that ∃d_{0 }> 0, λ_{0 }> 0, such that ∥u_{0}∥_{2 }< d_{0 }and
where the constant C depends only ∥u_{0}∥_{2}. This completes the Proof of Theorem 2.7.
Proof of Theorem 2.8 Multiplying the equation in (2.9) by u and integrating over Ω, we obtain
Since p < q, q + β < p + α, it follows from the Hölder inequality that
with μ_{2 }= q, μ_{1 }= μ  q, μ_{3 }= μ_{1}(1  p/q) and τ = N(μ q)(q + β)(q^{2 }+ Nβ)^{1 }≤ 2.
Then (5.7) becomes
This implies that ∃d_{0 }> 0, λ_{1 }> 0, such that ∥u_{0}∥_{2 }< d_{0 }and
This is the estimate (2.10) and we finish the Proof of Theorem 2.8.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
CC proposed the topic and the main ideas. The main results in this article were derived by CC. FY and ZY participated in the discussion of topic. All authors read and approved the final manuscript.
Acknowledgements
The authors wish to express their gratitude to the referees for useful comments and suggestions.
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