Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 14, 1-14; http://www.math.u-szeged.hu/ejqtde/

On the local integrability and boundedness of solutions to quasilinear parabolic systems^{∗}

Tiziana Giorgi^{†}  Mike O’Leary

September 6, 2004

Abstract We introduce a structure condition of parabolic type, which allows for the generalization to quasilinear parabolic systems of the known results of integrability, and boundedness of local solutions to singular and degenerate quasilinear parabolic equations.

1 Introduction

In this note, we investigate under which conditions it is possible to extend to systems the results of local integrability and local boundedness known to hold for solutions to a general class of degenerate and singular quasilinear parabolic equations. In particular, we show that the results presented by DiBenedetto in [1, Chp. VIII] are true for a larger class of problems, by providing conditions under which one can recover for weak solutions of quasilinear parabolic systems the work contained in [5, 6]. Fundamental to our approach is a new condition for the parabolicity of systems, which can be viewed as the extension of an analogous notion for parabolic equations, introduced in [1, Lemma 1.1 pg 19].

Generalizations of the results in [1, Chp. VIII] to initial-boundary value problems for systems have been proven in [7].

We study systems of the general form:

\frac{∂}{∂t}u_{i} − \frac{∂}{∂x_{j}} A_{ij}(x, t, u, ∇u) = B_{i}(x, t, u, ∇u)  (1)

for i = 1, 2, ..., n, and (x, t) ∈ Ω_{T} ≡ Ω × (0, T) with Ω ⊆ R^{N}; where we assume A_{ij} and B_{i} to be measurable functions in Ω × (0, T) × R^{n} × R^{Nn}, here i = 1, 2, ..., n; j = 1, 2, ..., N.

∗1991 Mathematics Subject Classifications: 35K40, 35K655.

Key words and phrases: Singular and degenerate quasilinear parabolic systems, local integrability, local boundedness

†Partially supported by the National Science Foundation-funded ADVANCE Institutional Transformation Program at NMSU, fund # NSF0123690

By a weak solution of (1), we mean a function u = (u_{1}, u_{2}, …, u_{n}) with u ∈ L_{∞,loc}(0, T; L_{2,loc}(Ω)) ∩ L_{p,loc}(0, T; W_{p,loc}^{1}(Ω)) for some p > 1, which verifies

∫∫_{Ω_{T}} \{−u_{i} \frac{∂ϕ_{i}}{∂t} + A_{ij}(x, t, u, ∇u)\frac{∂ϕ_{i}}{∂x_{j}} \} dxdt = ∫∫_{Ω_{T}} B_{i}(x, t, u, ∇u)ϕ_{i} dxdt (2)

for all ϕ = (ϕ_{1}, ϕ_{2}, …, ϕ_{n}) ∈ C_{0}^{∞}(Ω_{T}; R^{n}).

To the system (1), we add the following classical structure conditions (see [1, Chp. VIII]). For a.e. (x, t) ∈ Ω_{T}, every u ∈ R^{n}, and v ∈ R^{Nn}, we assume that

(H1) ∑_{j=1}^{N} ∑_{i=1}^{n} A_{ij}(x, t, u, v)v_{ij} ≥ C_{0}|v|^{p} − C_{3}|u|^{δ} − ϕ_{0}(x, t);

(H2) |A_{ij}(x, t, u, v)| ≤ C_{1}|v|^{p−1} + C_{4}|u|^{δ(1− \frac{1}{p})} + ϕ_{1}(x, t);

(H3) |B_{i}(x, t, u, v)| ≤ C_{2}|v|^{p(1− \frac{1}{δ})} + C_{5}|u|^{δ−1} + ϕ_{2}(x, t),

for C_{0} > 0, C_{1}, C_{2}, …, C_{5} ≥ 0, with δ s.t. 1 < p ≤ δ < (\frac{N + 2}{N}) p ≡ m, and

where ϕ_{0}, ϕ_{1}, ϕ_{2} are non-negative functions which satisfy

(H4) ϕ_{0} ∈ L_{1,loc}(Ω_{T}), ϕ_{1} ∈ L_{\frac{p}{p−1} ,loc}(Ω_{T}), and ϕ_{2} ∈ L_{\frac{m}{m−1} ,loc}(Ω_{T}).

Finally, we introduce and assume the parabolicity condition

(H5) ∑_{i,k=1}^{n} ∑_{j=1}^{N} A_{ij}(x, t, u, v)u_{i}u_{k}v_{kj} ≥ 0.

The main result of our work is the complete recovery for systems of the form (1) of Theorem 1 in [5]:

Theorem 1 Let u be a weak solution of (1), and suppose that the structure conditions (H1)-(H5) hold true, together with the following additional hypotheses:

(H6) ϕ_{0} ∈ L_{μ,loc}(Ω_{T}), ϕ_{1}, ϕ_{2} ∈ L_{s,loc}(Ω_{T}), where μ > 1 and s > \frac{(N + 2)p}{(N + 2)p − N};

(H7) u ∈ L_{r,loc}(Ω_{T}), with r > 1 and N(p − 2) + rp > 0.

If s, μ > \frac{(N+p)}{p}, then u ∈ L_{∞,loc}(Ω_{T});

if s = μ = \frac{(N+p)}{p}, then u ∈ L_{q,loc}(Ω_{T}) for any q < ∞;

if s, μ < \frac{(N+p)}{p}, then u ∈ L_{q,loc}(Ω_{T}) for any q < q^{∗}, where

q^{∗} = min \{\frac{s(Np + p − N)}{sN − (s − 1)(N + p)}, \frac{μ(Np + 2p)}{μN − (μ − 1)(N + p)}\}.

Remark We would like to point out that the parabolicity condition (H5) is a quite natural one to consider. In fact, for the case of a single equation it reduces to the condition

A_{j}(x, t, u, v)u^{2}v_{j} ≥ 0,

which, for u ≠ 0, is equivalent to the weak parabolicity condition presented in [1, Lemma 1.1, p.19].

Further, in the simple case where

\frac{∂u_{i}}{∂t} − \frac{∂}{∂x_{j}} (a_{jm}(x, t, u, ∇u)\frac{∂u_{i}}{∂x_{m}} ) = B_{i}(x, t, u, ∇u);

our requirement is satisfied if the matrix a_{jm}(x, t, u, ∇u) is for example positive definite. Indeed, since for the above system one has the identity

∑_{i,k=1}^{n} ∑_{j=1}^{N} A_{ij}(x, t, u, v)u_{i}u_{k}v_{kj} = ∑_{i,k=1}^{n} ∑_{j,m=1}^{N} a_{jm}(x, t, u, v) (u_{i}v_{im}) (u_{k}v_{kj}) ,

(H5) can be rewritten as

∑_{i,k=1}^{n} ∑_{j=1}^{N} A_{ij}(x, t, u, v)u_{i}u_{k}v_{kj} = ∑_{j,m=1}^{N} a_{jm}(x, t, u, v)w_{m}w_{j} ≥ 0,

where we set w_{h} = ∑_{l} u_{l}v_{lh}.

Finally, we note that (H5) is not so restrictive that the equation must have one of these simple forms. For example, consider the perturbation

A_{ij}(x, t, u, v) = a_{jm}(x, t, u, v)v_{im} + α_{ij}(x, t, u, v)

where the matrix a_{jm} is positive definite. Define

λ(x, t, u, v) = min_{|w|=1} a_{jm}(x, t, u, v)w_{j}w_{m} > 0;

this exists and is obtained because

w ↦ a_{jm}(x, t, u, v)w_{j}w_{m}

is positive and continuous for each (x, t, u, v) on the compact set {w ∈ R^{N} : |w| = 1}. Then for any vector w ∈ R^{N}, w ≠ 0

a_{jm}(x, t, u, v)w_{j}w_{m} = a_{jm} \frac{w_{j}}{|w|} \frac{w_{m}}{|w|} |w|^{2} ≥ λ|w|^{2}.

Condition (H5) will be verified if the perturbation α_{ij} satisfies the smallness condition

∑_{j=1}^{N} ∑_{i=1}^{n} |α_{ij}(x, t, u, v)u_{i}| ≤ λ(x, t, u, v) ∑_{j=1}^{N} |∑_{i=1}^{n} u_{i}v_{ij}| .

Indeed, we have

∑_{i,k=1}^{n} ∑_{j=1}^{N} A_{ij}(x, t, u, v)u_{i}u_{k}v_{kj} = ∑_{i,k=1}^{n} ∑_{j=1}^{N} [ ∑_{m=1}^{N} a_{jm}v_{im} + α_{ij}] u_{i}u_{k}v_{kj}

= ∑_{j,m=1}^{N} a_{jm} (∑_{i=1}^{n} u_{i}v_{im}) (∑_{k=1}^{n} u_{k}v_{kj}) + ∑_{j=1}^{n} (∑_{i=1}^{n} α_{ij}u_{i}) (∑_{k=1}^{n} u_{k}v_{kj})

≥ ∑_{j=1}^{N} {λ (∑_{k=1}^{n} u_{k}v_{kj})^{2} − |∑_{i=1}^{n} α_{ij}u_{i}| |∑_{k=1}^{n} u_{k}v_{kj}|} ≥ 0.

We follow the approach of [1, 5, 6] and start with the derivation, presented in Section 2, of a local energy estimates for weak solutions to (1). We then outline, in Section 3 and Section 4 how the methods in [5] can be applied to obtain local integrability and boundedness.

We also remark that the techniques presented can be modified to handle doubly degenerate problems, where A_{ij}(x, t, u, v)v_{ij} ≥ Φ(|u|)|v|^{p} − C_{3}|u|^{δ} − ϕ_{o}(x, t) for some Φ, following the same lines as the proof in [6].

2 Energy Estimates for u

2.1 Notation & Preliminaries

Let (x_{0}, t_{0}) ∈ Ω_{T}, without loss of generality we can assume (x_{0}, t_{0}) = (0, 0). For R > 0 we set Q_{R} = B_{R}(0) × (−R^{p}, 0), and for −R^{p} ≤ τ ≤ 0, we define Q_{R}^{τ} = B_{R}(0) × (−R^{p}, τ). For a fixed 0 < σ < 1, we consider a function ζ ∈ C^{∞}(Ω_{T}) with 0 ≤ ζ ≤ 1, ζ = 1 in Q_{σR}^{τ}, and ζ = 0 near |x| = R or t = −R^{p}. We also require that

|ζ_{t}| + |∇ζ|^{p} ≤ \frac{C_{σ}}{R^{p}} = \frac{2}{(1 − σ)^{p}R^{p}} .

We denote by ζ_{k} ∈ C_{0}^{∞}(Q_{R}^{τ}) the elements of a sequence of functions ζ_{k} → ζ uniformly in Q_{R}^{τ}. While, for η > 0, we let J_{η} be a smooth, symmetric, mollifying kernel in space-time, and for a given function f we use the notation f_{η} ≡ J_{η} ∗ f to represent its convolution with J_{η}.

Finally, for fixed ϵ > 0, and κ > 0, we consider the function

f(s) = \frac{(s − κ)_{+}}{(s − κ)_{+} + ϵ}.  (3)

In the following, we will use the fact that 0 ≤ f(s) ≤ 1, and that

f′(s) = \begin{cases}0 & s < κ, \\ \frac{ϵ}{[(s − κ)_{+} + ϵ]^{2}} & s > κ \end{cases}

verifies

0 ≤ f′(s) ≤ \begin{cases}0 & s < κ, \\ \frac{1}{ϵ} & κ < s < 2κ, \\ \frac{1}{s κ} & s > 2κ, \end{cases}  (4)

provided 0 < ϵ < \frac{1}{2}.

We are now ready to start the derivation of our energy estimate. Fix η > 0, κ > 0 and consider the test function {u_{i,η}(x, t)f(|u_{η}(x, t)|)ζ_{k}^{p}(x, t)}_{η} . Because this is a C_{0}^{∞} function for η sufficiently small, we can substitute it into the definition of weak solution to obtain

\begin{aligned}_{∫∫Ω_{T}} & − u_{i} \frac{∂}{∂t} \{u_{i,η}f(|u_{η}|)ζ_{k}^{p}\}_{η} dx dt \\ & + ∫∫_{Ω_{T}} A_{ij}(x, t, u, ∇u)\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ_{k}^{p}\}_{η} dx dt \\ = & ∫∫_{Ω_{T}} B_{i}(x, t, u, ∇u) \{u_{i,η}f(|u_{η}|)ζ_{k}^{p}\}_{η} dx dt. \end{aligned}  (5)

For convenience of notation, we rewrite (5) in compact form as I_{1} + I_{2} = I_{3}, and discuss each of these terms in turn.

2.2 Estimate of I_{1}

We begin by using the symmetry of the mollifying kernel, and integration by parts to rewrite I_{1} as

I_{1} = − ∫∫_{Ω_{T}} u_{i,η} \frac{∂}{∂t} \{u_{i,η}f(|u_{η}|)ζ_{k}^{p}\} dx dt

= ∫∫_{Q_{R}^{τ}} (\frac{∂}{∂t}u_{i,η}) u_{i,η}f(|u_{η}|)ζ_{k}^{p} dx dt.

We then notice that summing over the index i implies

∑_{i} u_{i,η} \frac{∂}{∂t}u_{i,η} = \frac{1}{2} ∑_{i} \frac{∂}{∂t} (u_{i,η})^{2} = \frac{1}{2} \frac{∂}{∂t}|u_{η}|^{2} = |u_{η}|\frac{∂}{∂t}|u_{η}|,  (6)

and we derive

I_{1} = ∫∫_{Q_{R}^{τ}} |u_{η}|\frac{∂|u_{η}|}{∂t}f(|u_{η}|)ζ_{k}^{p} dx dt.

If we now let k → ∞, thanks to the uniform convergence of ζ_{k} → ζ, and the smoothness of the mollified functions we obtain

lim_{k→∞} I_{1} = ∫∫_{Q_{R}^{τ}} |u_{η}|\frac{∂|u_{η}|}{∂t}f(|u_{η}|)ζ^{p} dx dt.

Proceeding in a standard fashion, we rewrite the integral on the right hand side as

∫∫_{Q_{R}^{τ}} \frac{∂}{∂t} (∫_{0}^{|u_{η}|} sf(s) ds) ζ^{p} dx dt

= ∫∫_{Q_{R}^{τ}} \frac{∂}{∂t} \{(∫_{0}^{|u_{η}|} sf(s) ds) ζ^{p}\} dx dt

− p ∫∫_{Q_{R}^{τ}} (∫_{0}^{|u_{η}|} sf(s) ds) ζ^{p−1}ζ_{t} dx dt,

and applying integration by parts, since ζ = 0 on t = −R^{p}, we gather

\begin{aligned}lim_{k→∞} I_{1} & = ∫_{B_{R}} (∫_{0}^{|u_{η}|} sf(s) ds) ζ^{p} dx|_{t=τ} \\ & − p ∫∫_{Q_{R}^{τ}} (∫_{0}^{|u_{η}|} sf(s) ds) ζ^{p−1}ζ_{t} dx dt. \end{aligned} (7)

We would like to take the limit for η ↓ 0 in (7), and we are able to do so, since from

|∫_{0}^{|u_{η}|} sf(s) ds − ∫_{0}^{|u|} sf(s) ds| = |∫_{|u|}^{|u_{η}|} sf(s) ds| ≤ γ_{1} ||u_{η}|^{2} − |u|^{2}| ,

with γ_{1} = \frac{1}{2}(max |f|), we can conclude

|∫_{B_{R}} (∫_{0}^{|u_{η}|} sf(s) ds − ∫_{0}^{|u|} sf(s) ds) ζ^{p} dx|_{t=τ} |

≤ γ_{1} ∫_{B_{R}} ||u_{η}|^{2} − |u|^{2}| dx|_{t=τ} ⟶^{η↓0} 0

for a.e. τ, and

|∫∫_{Q_{R}^{τ}} (∫_{0}^{|u_{η}|} sf(s) ds − ∫_{0}^{|u|} sf(s) ds) ζ^{p−1}ζ_{t} dx dt|

≤ γ_{2} ∫∫_{Q_{R}^{τ}} ||u_{η}|^{2} − |u|^{2}| dx dt ⟶^{η↓0} 0,

where γ_{2} is a constant that depends on σ, R and p. (Note that the above limits are zero due to the fact that u ∈ L_{∞,loc}(0, T; L_{2,loc}(Ω)).) In conclusion, we have the following estimate

lim_{η↓0} lim_{k→∞} I_{1} = ∫_{B_{R}} (∫_{0}^{|u|} sf(s) ds) ζ^{p} dx|_{t=τ}

− p ∫∫_{Q_{R}^{τ}} (∫_{0}^{|u|} sf(s) ds) ζ^{p−1}ζ_{t} dx dt. (8)

2.3 Estimate of I_{2}

We start as in Section 2.2, and use the symmetry of the mollifying kernel to rewrite I_{2}:

I_{2} = ∫∫_{Q_{R}^{τ}} A_{ij,η}(x, t, u, ∇u)\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ_{k}^{p}\} dx dt.

We then take the limit for k → ∞, and by the smoothness of the mollified functions we obtain

lim_{k→∞} I_{2} = ∫∫_{Q_{R}^{τ}} A_{ij,η}(x, t, u, ∇u)\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ^{p}\} dx dt.  (9)

As done while deriving the estimate for I_{1}, we would like to consider the limit for η ↓ 0 as well. To do so, we notice that the structure condition (H2) implies the inequality

∫∫_{Q_{R}^{τ}} |A_{ij}(x, t, u, ∇u)|^{\frac{p}{p−1}} dx dt ≤ γ ∫∫_{Q_{R}^{τ}} [|∇u|^{p} + |u|^{δ} + ϕ1^{\frac{p}{p−1}} ] dx dt.

From which, we have that A_{ij}(x, t, u, ∇u) ∈ L_{\frac{p}{p−1}} (Q_{R}^{τ}), since δ < m and since by the classical embedding theorems for parabolic spaces we know

u ∈ L_{∞,loc}(0, T; L_{2,loc}(Ω)) ∩ L_{p,loc}(0, T; W_{p,loc}^{1}(Ω)) ↪ L_{m,loc}(Ω_{T}).  (10)

Therefore, we obtain A_{ij,η}(x, t, u, ∇u) ⟶^{η↓0} A_{ij}(x, t, u, ∇u) in L_{\frac{p}{p−1}} (Q_{R}^{τ}).

On the other hand,

\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ^{p}\} = \frac{∂u_{i,η}}{∂x_{j}} f(|u_{η}|)ζ^{p}

+ u_{i,η}f′(|u_{η}|)\frac{∂|u_{η}|}{∂x_{j}}ζ^{p} + pu_{i,η}f(|u_{η}|)ζ^{p−1} \frac{∂ζ}{∂x_{j}};

hence from u_{i,η} → u_{i} and ∇u_{i,η} → ∇u_{i} almost everywhere [3, Appendix C, Theorem 6] we conclude that

\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ^{p}\} → \frac{∂}{∂x_{j}} \{u_{i}f(|u|)ζ^{p}\} a.e.

If next we use our estimates for f and f′, we have the upper bound

\begin{aligned}|\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ^{p}\}|^{p} & ≤ \{|∇u_{i,η}| + 2κ\frac{1}{ϵ}|∇u_{η}| + |u_{η}|\frac{1}{κ|u_{η}|}|∇u_{η}| + C|u_{η}|\}^{p} \\ & ≤ C \{|∇u_{η}|^{p} + |u_{η}|^{p}\} , \end{aligned}

which, applying a slight generalization of Lebesgue’s Dominated Convergence Theorem [4, §1.8], gives

\frac{∂}{∂x_{j}} \{u_{i,η}f(|u_{η}|)ζ^{p}\} ⟶^{η↓0} \frac{∂}{∂x_{j}} \{u_{i}f(|u|)ζ^{p}\} in L_{p}(Q_{R}^{τ}).

We then have that equation (9) yields

lim_{η↓0} lim_{k→∞} I_{2} = ∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)\frac{∂u_{i}}{∂x_{j}} f(|u|)ζ^{p} dx dt

+ ∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)u_{i}f′(|u|)\frac{∂|u|}{∂x_{j}}ζ^{p} dx dt

+ ∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)u_{i}f(|u|)p ζ^{p−1} \frac{∂ζ}{∂x_{j}} dx dt. (11)

The first integral above can be estimated with the help of (H1) as follows:

∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)\frac{∂u_{i}}{∂x_{j}} f(|u|)ζ^{p} dx dt ≥ C_{0} ∫∫_{Q_{R}^{τ}} |∇u|^{p}f(|u|)ζ^{p} dx dt

− C_{3} ∫∫_{Q_{R}^{τ}} |u|^{δ}f(|u|)ζ^{p} dx dt − ∫∫_{Q_{R}^{τ}} ϕ_{0}(x, t)f(|u|)ζ^{p} dx dt. (12)

To handle the second integral, we use the parabolicity assumption (H5), and the equal-

ity \frac{∂|u|}{∂x_{j}} = \frac{∂u_{k}}{∂x_{j}} \frac{u_{k}}{|u|}, true for u ≠ 0:

∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)u_{i}f′(|u|)\frac{∂|u|}{∂x_{j}}ζ^{p} dx dt

= ∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)u_{i}u_{k} \frac{∂u_{k}}{∂x_{j}} \frac{f′(|u|)}{|u|}ζ^{p} dx dt ≥ 0. (13)

For the last integral, we need (H2) to derive

∫∫_{Q_{R}^{τ}} A_{ij}(x, t, u, ∇u)u_{i}f(|u|)p ζ^{p−1} \frac{∂ζ}{∂x_{j}} dx dt

≥ −p C_{1} ∫∫_{Q_{R}^{τ}} |∇u|^{p−1}|u|f(|u|)ζ^{p−1}|∇ζ| dx dt

− p ∫∫_{Q_{R}^{τ}} (C_{4} |u|^{δ(1− \frac{1}{p})+1} f(|u|) ζ^{p−1}|∇ζ| + ϕ_{1}(x, t)|u|f(|u|)ζ^{p−1}|∇ζ|) dx dt. (14)

Finally, we combine (11), (12), (13), and (14) so to obtain the inequality:

lim_{η↓0} lim_{k→∞} I_{2} ≥ C_{0} ∫∫_{Q_{R}^{τ}} |∇u|^{p}f(|u|)ζ^{p} dx dt − C_{3} ∫∫_{Q_{R}^{τ}} |u|^{δ}f(|u|)ζ^{p} dx dt

− ∫∫_{Q_{R}^{τ}} ϕ_{0}(x, t)f(|u|)ζ^{p} dx dt − pC_{1} ∫∫_{Q_{R}^{τ}} |∇u|^{p−1}|u|f(|u|)ζ^{p−1}|∇ζ| dx dt

− p C_{4} ∫∫_{Q_{R}^{τ}} |u|^{δ(1− \frac{1}{p})+1}f(|u|)ζ^{p−1}|∇ζ| dx dt − p ∫∫_{Q_{R}^{τ}} ϕ_{1}(x, t)|u|f(|u|)ζ^{p−1}|∇ζ| dx dt. (15)

2.4 Estimate of I_{3}

Once again, our first step is to rewrite I_{3} in the form

I_{3} = ∫∫_{Q_{R}^{τ}} B_{i,η}(x, t, u, ∇u) {u_{i,η}f(|u_{η}|)ζ_{k}^{p}} dx dt,

and to consider the limit for k → ∞:

lim_{k→∞} I_{3} = ∫∫_{Q_{R}^{τ}} B_{i,η}(x, t, u, ∇u) {u_{i,η}f(|u_{η}|)ζ^{p}} dx dt.

To justify taking the limit for η ↓ 0 in this case, we proceed by noticing that (H3) implies

|B_{i}(x, t, u, ∇u)|^{\frac{m}{m−1}} ≤ C_{2}|∇u|^{p(1− \frac{1}{δ})(\frac{m}{m−1})} + C_{5}|u|^{m \frac{δ−1}{m−1}} + ϕ2^{\frac{m}{m−1}} (x, t).

Which yields B_{i,η} ⟶ B_{i} in L\frac{m}{m−1} (Q_{R}^{τ}), in view of the embedding (10), and the

relations δ < m, p (1 − \frac{1}{δ}) (\frac{m}{m−1}) = p (\frac{1−1/δ}{1−1/m}) < p. Moreover, since we know that

u_{i,η}f(|u_{η}|)ζ^{p} ⟶ u_{i}f(|u|)ζ^{p} for a.e. (x, t), and |u_{i,η}f(|u_{η}|)ζ^{p}|^{m} ≤ C|u_{η}|^{m};

we can apply the same generalization of Lebesgue’s Dominated Convergence Theorem to see that

u_{i,η}f(|u_{η}|)ζ^{p} ⟶ u_{i}f(|u|)ζ^{p} in L_{m}(Q_{R}^{τ}).

Thus,

lim_{η↓0} lim_{k→∞} I_{3} = ∫∫_{Q_{R}^{τ}} B_{i}(x, t, u, ∇u)u_{i}f(|u|)ζ^{p} dx dt,  (16)

and we can use (H3) once more to conclude

lim_{η↓0} lim_{k→∞} I_{3} ≤ C_{2} ∫∫_{Q_{R}^{τ}} |∇u|^{p(1− \frac{1}{δ})}|u|f(|u|)ζ^{p} dx dt

+ C_{5} ∫∫_{Q_{R}^{τ}} |u|^{δ}f(|u|)ζ^{p} dx dt + ∫∫_{Q_{R}^{τ}} ϕ_{2}(x, t)|u|f(|u|)ζ^{p} dx dt. (17)

2.5 The Energy Estimate

To derive our energy estimate (presented in Proposition 3 below), we use the intermediate result stated as Lemma 2. This is a direct consequence of equations (8), (15) and (17): starting from (5), one can use the bounds given in the previous sections, and then apply Young’s inequality to treat the terms involving |∇u|^{p−1}, |∇u|^{p(1− \frac{1}{δ})}, and |u|^{p(1− \frac{1}{δ})}.

Lemma 2 Let p > 1, let f be defined by (3), and let u ∈ L_{∞,loc}(0, T; L_{2,loc}(Ω)) ∩ L_{p,loc}(0, T; W_{p,loc}^{1}(Ω)) be a weak solution of (1). If the assumptions (H1)-(H5) are verified, then for any Q_{R}^{τ}(x_{0}, t_{0}) = B_{R}(x_{0}) × (t_{0} − R^{p}, τ) ⊂⊂ Ω_{T} we have

\begin{matrix}∫_{B_{R}} (∫_{0}^{|u|} sf(s) ds) ζ^{p} dx|_{t=τ} + ∫∫_{Q_{R}^{τ}} |∇u|^{p}f(|u|)ζ^{p} dx dt \\ ≤ γ (∫∫_{Q_{R}^{τ}} |u|^{δ}f(|u|)ζ^{p} dx dt + ∫∫_{Q_{R}^{τ}} |u|^{p}f(|u|)|∇ζ|^{p} dx dt \\ + ∫∫_{Q_{R}^{τ}} (∫_{0}^{|u|} sf(s) ds) ζ^{p−1}|ζ_{t}| dx dt + ∫∫_{Q_{R}^{τ}} ϕ_{0}(x, t)f(|u|)ζ^{p} dx dt+ \\ ∫∫_{Q_{R}^{τ}} ϕ_{1}(x, t)|u|f(|u|)ζ^{p−1}|∇ζ| dx dt + ∫∫_{Q_{R}^{τ}} ϕ_{2}(x, t)|u|f(|u|)ζ^{p} dx dt) \end{matrix} ; (18)

for some constant γ = γ(C_{0}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, p).

To extract useful information from Lemma 2, we need to substitute our choice of f(s), and then let ϵ ↓ 0. We first note that

∫_{0}^{|u|} sf(s) ds = ∫_{0}^{|u|} \frac{(s − κ)_{+}}{(s − κ)_{+} + ϵ} s ds ≥ ∫_{0}^{|u|} \frac{(s − κ)_{+}^{2}}{(s − κ)_{+} + ϵ} ds

≥ χ[|u| > κ] ∫_{κ}^{|u|} \frac{(s − κ)^{2}}{(s − κ) + ϵ} ds ≥ ∫_{0}^{(|u|−κ)_{+}} \frac{s^{2}}{s + ϵ} ds

≥ \frac{1}{2}(|u| − κ)_{+}^{2} − ϵ(|u| − κ)_{+} + ϵ^{2} ln [(|u| − κ)_{+} + ϵ] − ϵ^{2} ln ϵ, (19)

which implies

lim_{ϵ↓0} ∫_{0}^{|u|} sf(s) ds = ∫_{0}^{|u|} \frac{(s − κ)_{+}}{(s − κ)_{+} + ϵ} s ds ≥ \frac{1}{2}(|u| − κ)_{+}^{2};

and hence

lim_{ϵ↓0} ∫_{B_{R}} (∫_{0}^{|u|} sf(s) ds) ζ^{p} dx|_{t=τ} ≥ \frac{1}{2} ∫_{B_{R}}(|u| − κ)_{+}^{2}ζ^{p} dx|_{t=τ} .

By remarking that if ϵ < κ < s then \frac{(s − κ)s}{s − κ + ϵ} = s − ϵ [1 + \frac{κ − ϵ}{s − κ + ϵ}] ≤ s, one can

see that

\begin{aligned}∫_{0}^{|u|} sf(s) ds & = ∫_{0}^{|u|} \frac{(s − κ)_{+}}{(s − κ)_{+} + ϵ} s ds = χ[|u| > κ] ∫_{κ}^{|u|} \frac{(s − κ)s}{s − κ + ϵ} ds \\ & ≤ χ[|u| > κ] ∫_{κ}^{|u|} s ds ≤ \frac{1}{2}|u|^{2}χ[|u| > κ]. \end{aligned}

Moreover, since

f(|u|) = \frac{(|u| − κ)_{+}}{(|u| − κ)_{+} + ϵ} ↑ χ[|u| > κ] as ϵ ↓ 0,

we can use the Monotone Convergence Theorem to pass to the limit as ϵ ↓ 0 in the remaining terms of (18), and gather the bound

\frac{1}{2} ∫_{B_{R}}(|u| − κ)_{+}^{2}ζ^{p} dx|_{t=τ} + ∫∫_{Q_{R}^{τ}} |∇u|^{p}χ[|u| > κ]ζ^{p} dx dt

≤ γ (∫∫_{Q_{R}^{τ}} |u|^{δ}χ[|u| > κ] dx dt + ∫∫_{Q_{R}^{τ}} |u|^{p}χ[|u| > κ]|∇ζ|^{p} dx dt

+ ∫∫_{Q_{R}^{τ}} |u|^{2}χ[|u| > κ]ζ_{t} dx dt + ∫∫_{Q_{R}^{τ}} ϕ_{0}(x, t)χ[|u| > κ] dx dt

+ ∫∫_{Q_{R}^{τ}} ϕ_{1}(x, t)|u|χ[|u| > κ]|∇ζ| dx dt + ∫∫_{Q_{R}^{τ}} ϕ_{2}(x, t)|u|χ[|u| > κ] dx dt) .

In turn, the above inequality leads to the classical local energy estimate stated in Proposition 3 below, if one takes in account the relation |∇|u||^{p} ≤ |∇u|^{p}.

Proposition 3 (Local Energy Estimate) Under the hypotheses (H1)-(H5), if u is a weak solution of (1) then for Q_{R}(x_{0}, t_{0}) = B_{R}(x_{0}) × (t_{0} − R^{p}, t_{0}) ⊂⊂ Ω_{T}, 0 < σ < 1, and κ > 0

\begin{matrix}ess sup_{−R^{p}<τ<0} ∫_{B_{σR}}(|u| − κ)_{+}^{2} dx|_{t=τ} + ∫∫_{Q_{σR}} |∇(|u| − κ)_{+}|^{p} ζ^{p} dx dt \\ ≤ γ (∫∫_{Q_{R}} |u|^{δ}χ[|u| > κ] dx dt + \frac{1}{(1 − σ)^{p}R^{p}} ∫∫_{Q_{R}} |u|^{p}χ[|u| > κ] dx dt \\ + \frac{1}{(1 − σ)^{p}R^{p}} ∫∫_{Q_{R}} |u|^{2}χ[|u| > κ] dx dt + ∫∫_{Q_{R}} ϕ_{0}(x, t)χ[|u| > κ] dx dt \\ + \frac{1}{(1 − σ)R} ∫∫_{Q_{R}} ϕ_{1}(x, t)|u|χ[|u| > κ] dx dt \\ + ∫∫_{Q_{R}} ϕ_{2}(x, t)|u|χ[|u| > κ] dx dt) . \end{matrix} ; (20)

for some constant γ = γ(C_{0}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, p).

3 Higher Integrability of u

Owing to Proposition 3, we can proceed as in [5] to show higher integrability properties for u, that is the first part of Theorem 1. In fact, thanks to the Sobolev embedding for parabolic spaces [1, Chap. 1], and hypotheses (H6) for the functions ϕ_{0}, ϕ_{1}, and ϕ_{2}, we have

\begin{matrix}(∫∫_{Q_{σR}}(|u| − κ)_{+}^{p(\frac{N+2}{N})}ζ^{p} dx dt)^{\frac{1}{1+ \frac{p}{N}}} ≤ γ ∫∫_{Q_{R}} |u|^{δ}χ[|u| > κ] dx dt \\ + \frac{γ}{(1 − σ)^{p}R^{p}} ∫∫_{Q_{R}} |u|^{p}χ[|u| > κ] dx dt \\ + \frac{γ}{(1 − σ)^{p}R^{p}} ∫∫_{Q_{R}} |u|^{2}χ[|u| > κ] dx dt + γ ||ϕ_{0}||_{L_{μ}(Q_{R})}(meas[|u| > κ])^{1− \frac{1}{μ}} \\ + γ (\frac{||ϕ_{1}||_{L_{s}(Q_{R})}}{(1 − σ)R} + ||ϕ_{2}||_{L_{s}(Q_{R})}) ∫∫_{Q_{R}} |u|χ[|u| > κ] dx dt. \end{matrix} ; (21)

Inequality (21) is the key link needed to obtain for our systems exactly the same higher integrability result proven in [5, Proposition 3] for single equations:

Proposition 4 Under the hypotheses and notation of Theorem 1, we have that

if s, μ ≥ \frac{(N+p)}{p}, then u ∈ L_{q,loc}(Ω_{T}) for any q < ∞;

if s, μ < \frac{(N+p)}{p}, then u ∈ L_{q,loc}(Ω_{T}) for any q < q^{∗}.

Indeed, suppose u ∈ L_{β,loc}. Then we can use (21) to see that

\{κ^{(^{\frac{N+2}{N}})p} meas_{Q_{σR}}[|u| > 2κ]\}^{\frac{1}{1+p/N}}

≤ C_{γ,R,σ,p}||u||_{L_{β}(Q_{R})}^{β}\{ (\frac{1}{κ})^{β−δ} + (\frac{1}{κ})^{β−p} + (\frac{1}{κ})^{β−2} \}

+ C||u||_{L_{β}(Q_{R})}^{β(1− \frac{1}{μ})} (\frac{1}{κ})^{β(1− \frac{1}{μ})} + C||u||_{L_{β}(Q_{R})}^{β(1− \frac{1}{s})} (\frac{1}{κ})^{β(1− \frac{1}{s})−1} .

Therefore, if

α(β) = \frac{N + 2}{N}p + (1 + \frac{p}{N} ) min \{β − 2, β − δ, β (1 − \frac{1}{s}) − 1, β (1 − \frac{1}{μ})\}

then u ∈ L_{α(β),loc}^{weak}. Thus u ∈ L_{q,loc}(Q_{R}) for all q < α(β), and we can iterate this

process starting from β_{0} = max\{2, \frac{N + 2}{N}p, r\} to obtain the result. The details can be found in [5].

4 Boundedness of u

The L_{∞} local estimate part of Theorem 1 is a straightforward application of DeGiorgi’s technique; again the details can be found in [5]. In particular, we fix ρ > 0, σ > 0, so that Q_{ρ} ⊂⊂ Ω_{T}. For each integer n, we define

ρ_{n} = σρ + \frac{(1 − σ)}{2^{n}}ρ,

and set Q^{n} = Q_{ρ_{n}}. Next we fix κ > 0 to be chosen later, and set

κ_{n} = κ (1 − \frac{1}{2^{n+1}} ) .

For \frac{N+2}{N}p > 2, we consider

Y_{n} = \frac{1}{meas Q^{n}} ∫∫_{Q^{n}} |u − κ_{n}|^{m} dx dt,

while for \frac{N+2}{N}p ≤ 2, we take

Y_{n} = \frac{1}{meas Q^{n}} ∫∫_{Q^{n}} |u − κ_{n}|^{λ} dx dt,

for λ sufficiently large. This is well defined thanks to the local integrability proven in Section 3. We then apply the local energy estimate (21)in a standard way to obtain an estimate of the form

Y_{n+1} ≤ γ(B_{1}^{n}Y_{n}^{1+ϵ_{1}} + B_{2}^{n}Y_{n}^{1+ϵ_{2}} + B_{3}^{n}Y_{n}^{1+ϵ_{3}}),

for positive constants γ, B_{1}, B_{2}, B_{3}, ϵ_{1}, ϵ_{2} and ϵ_{3}. As final step, we choose κ sufficiently large so to have Y_{n} → 0 as n → ∞ which implies |u| < κ in Q_{σρ}.

It should be clear from the above presentation how the crucial roles in the generalization of the results in [5] to system of the form (1) are played by the local energy estimate of Proposition 3, and by the fact that the techniques in [5] really depend just on |u|. In a similar fashion, it is an easy exercise to check that the same ingredients (Proposition 3 and replacement of u by |u|) lead to the more general results of [6] .

References

[1] E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993.

[2] , Partial differential equations, Birkhäuser, Boston, 1995.

[3] L. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998.

[4] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, 1997.

[5] M. O’Leary, Integrability and boundedness of solutions to singular and degenerate quasilinear parabolic equations, Differential Integral Equations 12 (1999), 435– 452.

[6] , Integrability and boundedness for local solutions to doubly degenerate quasilinear parabolic equations, Adv. Differential Equations 5 (2000), no. 10-12, 1465–1492.

[7] W. H. Zajaczkowski, L_{∞}-Estimate for solutions of nonlinear parabolic systems, Singularities and differential equations (Warsaw, 1993), 491–501, Banach Center Publ., 33, Polish Acad. Sci., Warsaw, 1996.

Tiziana Giorgi Department of Mathematical Sciences, New Mexico State University Las Cruces, NM 88003 USA. E-mail address: tgiorgi@nmsu.edu

Mike O’Leary Mathematics Department, Towson University Towson, MD 21252 USA. E-mail address: moleary@towson.edu

(Received October 1, 2003)