### 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
*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 solution

### 1 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 so-called 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

for ∀*t *∈ [0, *T*] and
*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 [2-6] 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 Leray-Lions type problem

where *a*(*x, t, s, ξ *) is a Carath*é*odory 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 [8-10]. Since the equation in (1.1) contains the nonlinear gradient term *u*|*u*|^{α-2}|∇*u*|^{p }and *u*|*u*|^{β-2}|∇*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}) 2*q *≤ 2 + *m, α, β *< 2 + *m*(1 + 1/*N*)/2,

(*H*_{5}) the mean curvature of *H*(*x*) of *∂*Ω at *x *is non-positive 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

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 *+ 2*m *+ 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 + 2*N*^{-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

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

where *C *= *C*(∥*u*_{0}∥_{2}).

To obtain the above results, we will need the following Lemmas.

**Lemma 2.9 **(Gagliardo-Nirenberg 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 well-known Gagliardo-Nirenberg-Sobolev 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, λ*θ *≥ 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
*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*|^{k-1}*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,
*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 left-hand 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+β-1) }in

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

with*C*_{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*^{k-1}, *k *≥ 2, we have

It follows from the H*ö*lder and Sobolev inequalities that

in which *θ*_{1 }= *k*λ(*m *- *ν *+ (*m *+ 2)*N*^{-1}), *θ*_{2 }= *ν*λ(*m *+ 2)*N*^{-1}, *θ*_{3 }= *ν*λ(*k *+ *m*), *σ *= *ν*λ, *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 *μ*λ_{0}*θ *= 1, (1 - *μ*λ_{0})*θ' *= 1.

Note that *μ*_{1}*θ' *< *k*. Let

with *γ *= *qθ'θ*^{-1 }= *qμ*λ_{0}/(1 - *μ*λ_{0}). Then, (3.14) becomes

or

with *σ*_{0 }= max{*σ, γ*} = max{*ν*λ*, 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*^{n-2}(*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

with

Then, inserting (3.18) into (3.17) (*k *= *k*_{n}), we find that

or

where

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
*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(β-1)}(*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

Since

for some
*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*|^{k-2}∇*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 2*q *≤ *m *+ 2, 2*p *≤ *m *+ 2, then (4.1) becomes

where *h*_{1}(*t*) = max{*h*^{2(α-1)}(*t*), *h*^{2(β-1)}(*t*), *h ^{ν}*(

*t*)}. Since

*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*^{n-2 }(*R*-1-*m*) + *m *(*R*-1)^{-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 *+ 2*m *+ 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

with

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

with

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 H*ö*lder inequality that