Journal of Convex Analysis Volume 7 (2000), No. 1, 203–208

Summability of the Solutions to Nonlinear Parabolic Equations with Measure Data^{∗}

Li Feng-Quan Li Guang-wei

Department of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, China. e-mail: fqli@qfnu.edu.cn

Received July 23, 1999 Revised manuscript received December 18, 1999

In this paper, we give an improvement of the summability of solutions to nonlinear parabolic equations with measure data.

Keywords: summability, nonlinear parabolic equations, measure data

1991 Mathematics Subject Classification: 35D10, 35K60

1. Introduction

Let Ω be a bounded domain in R^{N}, N ≥ 2. For T > 0, let us denote by Q the cylinder Ω × (0, T), and by Γ the lateral surface ∂Ω × (0, T). Let p be a real number such that p > 2 − 1/(N + 1).

We consider the following nonlinear parabolic Cauchy-Dirichlet problem:

\begin{cases}\frac{∂u}{∂t} − div(a(x, t, u, Du)) = f & in Q, \\ u(x, t) = 0 & on Γ, \\ u(x, 0) = u_{0} & in Ω. \end{cases}  (P)

Here f and u_{0} belong to M(Q) and M(Ω), the space of bounded Radon measures on Q and Ω respectively. The function a(x, t, s, ξ) : Q × R × R^{N} → R^{N} is a Carathéodory function (i.e., it is continuous with respect to s and ξ for almost every (x, t) ∈ Q, and measurable with respect to (x, t) for every s ∈ R and ξ ∈ R^{N}). We assume that there exist two real positive constants λ_{0}, α and a positive function h ∈ L^{p′}(Q), such that for any s ∈ R, ξ ∈ R^{N}, η ∈ R^{N}, ξ ≠ η and for almost every (x, t) ∈ Q,

a(x, t, s, ξ)ξ ≥ λ_{0}|ξ|^{p}  (1.1)
|a(x, t, s, ξ)| ≤ α(h(x, t) + |s|^{p−1} + |ξ|^{p−1})  (1.2)
[a(x, t, s, ξ) − a(x, t, s, η)][ξ − η] > 0.  (1.3)

We look for weak solutions, i.e., for function u such that u ∈ L^{1}(0, T; W_{0}^{1,1}(Ω)), a(x, t, u, Du) ∈ L^{1}(Q) and

− ∫_{Q} u\frac{∂ϕ}{∂t}dxdt + ∫_{Q} a(x, t, u, Du)Dϕdxdt = ∫_{Q} ϕdf + ∫_{Ω} ϕ(x, 0)du_{0}(x),  (1.4)

∗This work is supported by NSF of Shandong (Y98A090/2, Q99A05).

ISSN 0944-6532 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

204 Li Feng-Quan, Li Guang-wei / Summability of the solutions to parabolic equations

for every ϕ ∈ C^{∞}(Q̄) which is zero in a neighborhood of Γ ∪ (Ω × {T }).

We recall the following results(see [1], [2] or [3]).

Theorem 1.1. Assume (1.1)–(1.3) hold, p > 2 − 1/(N + 1), f ∈ M(Q), u_{0} ∈ M(Ω). Then there exists a solution u of problem (P) such that

u ∈ L^{q}(0, T; W_{0}^{1,q}(Ω)), ∀q < p − \frac{N}{N + 1} = q_{0}.

Remark 1.2. We observe that q_{0} > 1 if and only if p > 2 − \frac{1}{N+1}.

In order to obtain u ∈ L^{q_{0}}(0, T; W_{0}^{1,q_{0}}(Ω)), a sufficient condition is given by the following theorem.

Theorem 1.3. Assume (1.1)–(1.3) hold, p > 2 − \frac{1}{N+1}, f ∈ L^{1}(0, T; L^{1} log L^{1}(Ω)), u_{0} ∈ L^{1} log L^{1}(Ω), where L^{1} log L^{1}(Ω) is the Orlicz space generated by the function s log(1 + s). Then there exists a solution u of problem (P) such that

u ∈ L^{q_{0}}(0, T; W_{0}^{1,q_{0}}(Ω)).

Remark 1.4. The results given in Theorem 1.1 and Theorem 1.3 are only proved in the case p > 2 − \frac{1}{N+1}. However, in general, if 1 < p ≤ 2 − \frac{1}{N+1}, f and u_{0} are only bounded measures(or even integrable functions), a weak solution in the sense of (1.4) doesn’t exist. Recently, existence of so-called renormalized solutions has been proved for problems of type (P) and of the corresponding elliptic problem for every p > 1 (see [4],[5]).

This paper extends the analogous result of [6] to parabolic equations. Here we give the improvement of the summability of the solution u to problem (P) only under the assumptions of Theorem 1.1.

2. Regularity results

In this section, we give an improvement of the summability of the solution u to problem (P). We begin by recalling the well-known Gagliardo-Nirenberg embedding theorem (see [8]).

Lemma 2.1. Let v be a function in W_{0}^{1,q}(Ω) ∩ L^{ρ}(Ω), with q ≥ 1, ρ ≥ 1. Then there exists a positive constant C_{1} depending on N, q, and ρ, such that

∥v∥_{L^{γ}(Ω)} ≤ C_{1}∥Dv∥_{L^{q}(Ω)}^{θ}∥v∥_{L^{ρ}(Ω)}^{1−θ},  (2.1)

for every θ and γ satisfying

0 ≤ θ ≤ 1, 1 ≤ γ < +∞, \frac{1}{γ} = θ(\frac{1}{q} − \frac{1}{N}) + \frac{1 − θ}{ρ}.  (2.2)

We state the main result of this paper.

Theorem 2.2. The solution u of problem (P) given by Theorem 1.1 is such that for every β > \frac{1}{p−1},

\frac{u}{[log(2 + |u|)]^{β}} ∈ L^{q_{0}}(0, T; W_{0}^{1,q_{0}}(Ω)),  (2.3)

Proof. We follow the method of [1], [2] or [3]. Let {f_{n}} ⊂ L^{∞}(Q), {u_{0n}} ⊂ L^{∞}(Ω), such that

f_{n} ⟶ f weak* in M(Q),  (2.4)
u_{0n} ⟶ u_{0} weak* in M(Ω).  (2.5)

Moreover, there exists a positive constant C_{2} independent of n, such that for every n

∥f_{n}∥_{L^{1}(Q)} ≤ C_{2},  (2.6)
∥u_{0n}∥_{L^{1}(Ω)} ≤ C_{2}.  (2.7)

We consider the approximate problem:

\begin{cases}\frac{∂u_{n}}{∂t} − div(a(x, t, u_{n}, Du_{n})) = f_{n} & in Q, \\ u_{n}(x, t) = 0 & on Γ, \\ u_{n}(x, 0) = u_{0n} & in Ω. \end{cases}  (P_{n})

For any given positive integer n, by [9], problem (P_{n}) admits a solution u_{n} ∈ L^{p}(0, T; W_{0}^{1,p}(Ω)) ∩ C([0, T]; L^{2}(Ω)) and satisfies

⟨u_{nt}, ψ⟩ + ∫_{Q} a(x, t, u_{n}, Du_{n})Dψdxdt = ⟨f_{n}, ψ⟩, ∀ψ ∈ L^{p}(0, T; W_{0}^{1,p}(Ω)).  (2.8)

For any given constant k > 0, we define the cut function T_{k} : R → R as

T_{k}(s) = \begin{cases}s & if |s| ≤ k, \\ k sign(s) & if |s| > k. \end{cases}  (2.9)

and let Θ_{k}(σ) be

Θ_{k}(σ) = ∫_{0}^{σ} T_{k}(s)ds.  (2.10)

For any given τ ∈ (0, T), we take ψ = T_{1}(u_{n})χ_{(0,τ)} in (2.8), where χ_{(0,τ)} denotes the characteristic function of set (0, τ) in [0, T], we easily obtain

∫_{Ω} Θ_{1}(u_{n}(x, τ))dx − ∫_{Ω} Θ_{1}(u_{0n})dx ≤ ∫_{Q} |f_{n}∥T_{1}(u_{n})|dxdt ≤ ∥f_{n}∥_{L^{1}(Q)}.  (2.11)

Since we have |s| − 1/2 ≤ Θ_{1}(s) ≤ |s| for every s ∈ R, we obtain for every τ ∈ (0, T),

∫_{Ω} |u_{n}(x, τ)|dx ≤ ∥f_{n}∥_{L^{1}(Q)} + 1/2measΩ + ∥u_{0n}∥_{L^{1}(Ω)}.  (2.12)

By (2.6), (2.12) yields

∥u_{n}∥_{L^{∞}(0,T;L^{1}(Ω))} ≤ C_{3},  (2.13)

where C_{3} is a positive constant independent of n.

For any given k ≥ 0, let us define the function ϕ_{k}(s) = min{(|s| − k)^{+}, 1} sign(s) and the set B_{k} = {(x, t) ∈ Q : k ≤ |u_{n}(x, t)| < k + 1}. We take ψ_{k} = ϕ_{k}(u_{n}) in (2.8), by (1.1), we obtain

∫_{Ω} ψ_{k}(u_{n}(x, T))dx − ∫_{Ω} ψ_{k}(u_{0n})dx + λ_{0} ∫_{B_{k}} |Du_{n}|^{p}dxdt ≤ ∥f_{n}∥_{L^{1}(Q)}.  (2.14)

where we have defined ψ_{k}(σ) = ∫_{0}^{σ} ϕ_{k}(s)ds. Since 0 ≤ ψ_{k}(σ) ≤ |σ|, by (2.6),

∫_{B_{k}} |Du_{n}|^{p}dxdt ≤ C_{4}  (2.15)

From now we denote by C_{4}, C_{5}, ⋯ various positive constants which only depend on the known data of the problem, but independent of n, u_{n}, k.

For any given β > \frac{1}{p−1}, let us define a function ϕ(s) as

ϕ(s) = \frac{s}{[log(2 + |s|)]^{β}} , ∀s ∈ R.  (2.16)

(2.13) and (2.16) imply that

∥ϕ(u_{n})∥_{L^{∞}(0,T;L^{1}(Ω))} ≤ C_{5}.  (2.17)

By (2.15) and β > \frac{1}{p−1}, we obtain

∫_{Q} \frac{|Du_{n}|^{p}}{(2 + |u_{n}|)[log(2 + |u_{n}|)]^{(p−1)β}} dxdt

= ∑_{k=0}^{∞} ∫_{B_{k}} \frac{|Du_{n}|^{p}}{(2 + |u_{n}|)[log(2 + |u_{n}|)]^{(p−1)β}} dxdt

≤ C_{4} ∑_{k=0}^{∞} \frac{1}{(2 + k)[log(2 + k)]^{(p−1)β}} ≤ C_{6}.

(2.18)

Setting g(u_{n}) = (2 + |u_{n}|)[log(2 + |u_{n}|)]^{(p−1)β}, Hölder’s inequality and (2.18) yield

\begin{aligned}∫_{Q} |Dϕ(u_{n})|^{q_{0}}dxdt & ≤ ∫_{Q} |Du_{n}|^{q_{0}}|ϕ′(u_{n})|^{q_{0}}dxdt \\ & = ∫_{Q} \frac{|Du_{n}|^{q_{0}}}{g(u_{n})^{q_{0}/p}} |ϕ′(u_{n})|^{q_{0}}g(u_{n})^{q_{0}/p}dxdt \\ & ≤ (∫_{Q} \frac{|Du_{n}|^{p}}{g(u_{n})} dxdt)^{\frac{q_{0}}{p}} (∫_{Q} |ϕ′(u_{n})|^{\frac{pq_{0}}{p−q_{0}}} g(u_{n})^{\frac{q_{0}}{p−q_{0}}} dxdt)^{1− \frac{q_{0}}{p}} \\ & ≤ C6^{\frac{q_{0}}{p}} (∫_{Q} |ϕ′(u_{n})|^{\frac{pq_{0}}{p−q_{0}}} g(u_{n})^{\frac{q_{0}}{p−q_{0}}} dxdt)^{1− \frac{q_{0}}{p}} \\ & ≤ C_{7}(∫_{Q}(1 + |ϕ(u_{n})|)^{\frac{q_{0}}{p−q_{0}}} dxdt)^{1− \frac{q_{0}}{p}} \\ & ≤ C_{8} + C_{9}(∫_{0}^{T} ∥ϕ(u_{n}(t))∥L^{\frac{q_{0}}{p−q_{0}}} (Ω)^{\frac{q_{0}}{p−q_{0}}}dt)^{1− \frac{q_{0}}{p}} . \end{aligned}  (2.19)

In fact, since ∀s ∈ R, (2 + |s|) log(2 + |s|) > 1 + |s|, then |ϕ′(s)| ≤ (1 + β)[log(2 + |s|)]^{−β}. Thus we have

|ϕ′(s)|^{p}g(s) ≤ (1 + β)^{p}[log(2 + |s|)]^{−β}(2 + |s|) ≤ 3(1 + β)^{p}[log 2)]^{−β}(1 + |ϕ(s)|).

The above inequality yields the first of these two last inequalities in (2.19).

Taking v = ϕ(u_{n}(t)) for a. e. t ∈ (0, T), q = q_{0}, ρ = 1, γ = \frac{q_{0}}{p−q_{0}} in Lemma 2.1, we have

∥ϕ(u_{n}(t))∥_{L^{\frac{q_{0}}{p−q_{0}}} (Ω)} ≤ C_{1}∥Dϕ(u_{n}(t))∥_{L^{q_{0}}(Ω)}^{θ}∥ϕ(u_{n}(t))∥_{L^{1}(Ω)}^{θ}, a. e. t ∈ (0, T),  (2.20)

and

\frac{p − q_{0}}{q_{0}} = θ(\frac{1}{q_{0}} − \frac{1}{N}) + 1 − θ, 0 ≤ θ ≤ 1.  (2.21)

(2.20), (2.17) and (2.19) imply that

∫_{Q} |Dϕ(u_{n})|^{q_{0}}dxdt ≤ C_{8} + C_{10}(∫_{0}^{T} ∥Dϕ(u_{n}(t))∥L^{q_{0}}(Ω)^{\frac{θq_{0}}{p−q_{0}}}dt)^{1− \frac{q_{0}}{p}} .  (2.22)

If we choose \frac{θ}{p−q_{0}} = 1, by (2.21), we must take θ = \frac{N}{N+1}. Thus we have

∫_{Q} |Dϕ(u_{n})|^{q_{0}}dxdt ≤ C_{8} + C_{10}(∫_{Q} |Dϕ(u_{n})|^{q_{0}}dxdt)^{1− \frac{q_{0}}{p}} .  (2.23)

Since 0 < 1 − \frac{q_{0}}{p} < 1, (2.23) yields

∫_{Q} |Dψ(u_{n})|^{q_{0}}dxdt ≤ C_{11}.  (2.24)

Poincaré’s inequality and (2.24) imply that

∫_{0}^{T} ∥ϕ(u_{n}(t))∥_{W_{0}^{1,q_{0}}(Ω)}^{q_{0}}dt ≤ C_{12}.  (2.25)

The method of [2] allow us to pass to the limit in the approximate equations and we can prove that there exists a solution u to problem (P) such that (2.3) hold. This completes the proof of Theorem 2.2.

Acknowledgements. The authors would like to thank the referee for his comments and suggestions.

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