A note on strongly nonlinear parabolic variational inequalities

Consider the parabolic initial-boundary value problem

on a cylinder Q_T = Ω × ]0, T[ over a bounded smooth domain Ω ⊂ ℝ^N, where

A(u)=∑|α|≤m(−1)|α|DαAα(x,t;Du(x,t)),G(u)=g(x,t;u)$$ \begin{equation} A(u)=\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}A_{\alpha}(x,t;Du(x,t)),\quad G(u)=g(x,t;u) \end{equation} $$(1)

and Du = (D^αu)_∣α∣≤m. If the coefficients A_α satisfy at most polynomial growth conditions in u and its space derivatives while g obeys no growth in u, but merely a sign condition, Landes and Mustonen [6] proved that the usual truncation can be utilized to obtain weak solutions of (1) when m = 1. In [1], Brézis and Browder considered (1) but under stronger hypotheses on g. Roughly speaking, they required g to be controlled from above and below by the derivative of some convex function. In [5], Landes proved that this assumption is not necessary provided a certain a priori bound for the time derivative of solutions were needed. In [2], Browder and Breézis established an existence and uniqueness result for a general class of variational inequalities for (1) when g obeys no growth condition while A is a regular elliptic operator. Their proof is based on a type of compactness result. In this note, we extend the result of [5] to the corresponding class of variational inequalities under weaker assumptions.

We start by assuming the following hypotheses.

The function spaces we shall deal with will be obtained by the completion of the space of smooth functions with respect to the appropriate norm. We denote by

X=Lp(0,T;W0m,p(Ω))=C1(0,T,C0∞(Ω))¯‖‖p;m,p,Lp(QT)=C0∞(Ω)¯‖‖p;p,$$ X=L^p(0,T;W_0^{m,p}(\Omega))=\mathcal{C}^1\overline{(0,T,\mathcal{C}_0^{\infty}(\Omega))}^{\|\ \|_{p;m,p}},\qquad L^p(Q_T)=\overline{\mathcal{C}_0^{\infty}(\Omega)}^{\|\ \|_{p;p}}, $$

where

‖u‖p;m,pp=∫0T‖u‖m,ppdt=∫0T∑|α|≤m‖Dαu‖ppdt‖u‖p;p=∫0T‖u‖PPdt and ‖u‖PP=∫Ω|u|pdx.$$ \begin{align*} \|u\|^p_{p;m,p}&=\int_0^T\|u\|^p_{m,p}dt=\int_0^T\sum_{|\alpha|\leq m}\|D^{\alpha}u\|^p_p dt\\ \|u\|_{p;p}&=\int_0^T\|u\|^P_P\, dt \ \text{ and } \ \|u\|_P^P=\int_{\Omega}|u|^p dx. \end{align*} $$

Put W = X ∩ C (0, T;L²(Ω)). Finally, we choose a sequence (Φi)i=1∞⊂C0∞(Ω) $(\Phi_i)_{i=1}^{\infty}\subset \mathcal{C}_0^{\infty}(\Omega)$ such that ∪n=1∞Vn $\cup_{n=1}^{\infty}V_n$ with V_n = span(Φ₁, Φ₂, …, Φ_n) is dense in W^{j, p}(Ω):jp > mp + N.

Denote by Y_n = C (0, T;V_n). Since the closure of ∪n=1∞Yn $\cup_{n=1}^{\infty}Y_n$ with respect to the C^m−topology contains C0∞(QT) $\mathcal{C}_0^{\infty}(Q_T)$ , then for f ∈ L²(Q_T) there exists fk∈∪n=1∞Yn $f_k\in \cup_{n=1}^{\infty}Y_n$ such that f_k → f in L²(Q_T) [4]. For simplicity, we fix the constant c throughout this note. Now we are in a position to give our result.

gk(x,t;u)={kg(x,t;u)|g(x,t;u)|if|g(x,t;u)|>kg(x,t;u)otherwise$$ g_k(x,t;u)= \begin{cases} k\frac{g(x,t;u)}{|g(x,t;u)|} & \text{ if } |g(x,t;u)|\gt k\\ g(x,t;u) & \text{otherwise.} \end{cases}$$

Consider the truncated variational inequality

〈u˙,v−u〉+〈A(u),v−u〉+∫QTgk(x,t,u)(v−u)dxdt≥〈f˜,v−u〉 for allv∈W∩K.$$ \begin{equation} \langle \dot{u},v-u\rangle+\langle A(u),v-u\rangle+\int_{Q_T}g_k(x,t,u)(v-u)\, dxdt\geq \langle \tilde{f},v-u\rangle \text{for all} v\in W\cap K. \end{equation} $$(3)

Firstly, we show the existence of a Galerkin solution u_ε ∈ Y_n∩ K of (3) with

‖u˙n‖2;2+‖un‖2;2+‖un‖p;m,p≤c,$$ \begin{equation} \|\dot{u}_n\|_{2;2}+\|u_n\|_{2;2}+\|u_n\|_{p;m,p}\leq c, \end{equation} $$(4)

where c is a constant not depending on ε, k, and n.

For this aim, let A_α,ε, g(α, ε), and f˜˜ϵ $\tilde{\tilde{f}}_{\epsilon}$ be the Friedrich’s mollification in the variables (x, t) ∈ ℝ^{N + 1} of A_α, g_k, and f˜ $\tilde{f}$ , respectively. There exists a Galerkin solution u_ϵ ∈ Y_n∩ K for the mollified variational inequality

∫0τ(u˙ε,v−uε)dt+∫0τAε(uε,v−uε)dt+∫QTgk,ε(x,t,uε)(v−uε)dxdt≥∫QTf˜ε(v−uε)dxdt$$ \begin{equation} \int_0^{\tau}(\dot{u}_{\epsilon},v-u_{\epsilon})dt+\int_0^{\tau} A_{\epsilon}(u_{\epsilon},v-u_{\epsilon})dt+\int_{Q_T}g_{k,\epsilon}(x,t,u_{\epsilon})(v-u_{\epsilon})\, dxdt\geq \int_{Q_T} \tilde{f}_{\epsilon}(v-u_{\epsilon})\, dxdt \end{equation} $$(5)

for all v ∈ Y_n∩ K and all τ ∈ ]0, T[ with

‖uε(t)‖2≤c$$ \|u_{\epsilon}(t)\|_2\leq c $$

(see [3, 4]). Put v = 0 in (5). Then we get from A₃ and G the estimate

‖uε‖p;p≤c.$$ \|u_{\epsilon}\|_{p;p}\leq c. $$

On the other hand, given h > 0, n ∈ ℕ, and any w_ε ∈ Y_n∩ K, put v = u_ε−hw_ε in (5). Then we get

∫0T(u˙ε,wε)dt+∫0T(Aε(uε),wε)dt+∫QTgk,ε(x,t,ueps)wεdxdt≤∫QTfεwεdxdt.$$ \int_0^T (\dot{u}_{\epsilon},w_{\epsilon})dt+\int_0^T (A_{\epsilon}(u_{\epsilon}),w_{\epsilon})dt+\int_{Q_T}g_{k,\epsilon}(x,t,u_{eps})w_{\epsilon}\, dxdt\leq \int_{Q_T}f_{\epsilon}w_{\epsilon}\, dxdt.$$

In particular,

∫0T(u˙ε(t),uε(t+h)−uε(t)h)dt+∫0T(Aε(uε),uε(t+h)−uε(t)h)dt+∫QTgk,ε(x,t,uε(t))(uε(t+h)−uε(t)h)dxdt≤∫0T(f˜ε(t),uε(t+h)−uε(t)h)dt.$$ \begin{align*} \int_0^T \left(\dot{u}_{\epsilon}(t),\frac{u_{\epsilon}(t+h)-u_{\epsilon}(t)}{h}\right)dt&+\int_0^T \left(A_{\epsilon}(u_{\epsilon}),\frac{u_{\epsilon}(t+h)-u_{\epsilon}(t)}{h}\right)dt\\ &+\int_{Q_T} g_{k,\epsilon}(x,t,u_{\epsilon}(t))\left(\frac{u_{\epsilon}(t+h)-u_{\epsilon}(t)}{h}\right) dxdt\\ &\leq \int_0^T (\tilde{f}_{\epsilon}(t),\frac{u_{\epsilon}(t+h)-u_{\epsilon}(t)}{h})dt. \end{align*} $$

Allowing h → 0, keeping ε fixed, we have

∫0T(u˙ε(t),u˙ε(t))dt+∫0T(Aε(uε),u˙ε(t))dt+∫QTgk,ε(x,t,uε)u˙ε(t)dxdt≤∫0T(f˜ε(t),u˙ε(t))dt.$$ \begin{align*} \int_0^T (\dot{u}_{\epsilon}(t),\dot{u}_{\epsilon}(t))dt&+\int_0^T (A_{\epsilon}(u_{\epsilon}),\dot{u}_{\epsilon}(t))dt+\int_{Q_T}g_{k,\epsilon}(x,t,u_{\epsilon})\dot{u}_{\epsilon}(t)dxdt\\ &\leq \int_0^T (\tilde{f}_{\epsilon}(t),\dot{u}_{\epsilon}(t))dt. \end{align*} $$

By A₄ and G, we obtain

∫0T‖u˙ε(t)‖22dt+∫0T∂∂t[Fε(x,t;Duε(t))+Γk,ε(x,t;uε(t))]dxdt≤c∫0T‖u˙ε(t)‖22dt.$$ \begin{align*} \int_0^T \|\dot{u}_{\epsilon}(t)\|_2^2\, dt&+\int_0^T \frac{\partial}{\partial t}[F_{\epsilon}(x,t; Du_{\epsilon}(t))+\Gamma_{k,\epsilon}(x,t; u_{\epsilon}(t))]\, dxdt\leq c\int_0^T \|\dot{u}_{\epsilon}(t)\|_2^2\, dt. \end{align*} $$

We may write this inequality in the form

∫0T‖u˙ε(t)‖22dt+32∫0T∂∂t∫Ω[Fε(x,t;Du(t))+Γk,ε(x,t;uε(t))]dxdt≤12∫0T∂∂t∫Ω[Fε(x,t;Duε(t))+Γk,ε(x,t;uε(t))]dxdt+c∫0T‖u˙ε(t)‖22dt.$$ \begin{align*} \int_0^T \|\dot{u}_{\epsilon}(t)\|_2^2\, dt&+\frac 32\int_0^T \frac{\partial}{\partial t}\int_{\Omega}[F_{\epsilon}(x,t;Du_{\epsilon}(t))+\Gamma_{k,\epsilon}(x,t;u_{\epsilon}(t))]\, dxdt\\ &\leq \frac 12\int_0^T \frac{\partial}{\partial t}\int_{\Omega} [F_{\epsilon}(x,t;Du_{\epsilon}(t))+\Gamma_{k,\epsilon}(x,t;u_{\epsilon}(t))]\, dxdt+c\int_0^T\|\dot{u}_{\epsilon}(t)\|_2^2\, dt. \end{align*} $$

From the mean value theorem for definite integrals, we get

∫0T‖u˙ϵ(t)‖22dt+cT∫Ω[∂∂tFϵ(x,t;Duϵ(t))+∂∂tΓk,ϵ(x,t;uϵ(t))]dx≤c+c∫0T∫Ω[∂∂tFϵ(x,t;Duϵ(t))+∂∂tΓk,ϵ(x,t;uϵ(t))]dxdt,0<t<T,$$ \begin{align*} \int_0^T \|\dot{u}_{\epsilon}(t)\|_2^2\, dt&+cT\int_{\Omega}\left[\frac{\partial}{\partial t}F_{\epsilon}(x,t;Du_{\epsilon}(t))+\frac{\partial}{\partial t}\Gamma_{k,\epsilon}(x,t;u_{\epsilon}(t))\right]\, dx\\ &\leq c+c\int_0^T \int_{\Omega}\left[\frac{\partial}{\partial t}F_{\epsilon}(x,t; Du_{\epsilon}(t))+\frac{\partial}{\partial t}\Gamma_{k,\epsilon}(x,t;u_{\epsilon}(t))\right]\, dxdt, \quad 0\lt t\lt T, \end{align*} $$

where

Γk,ε(x,t;ρ)=∫0ρgk,ε(x,t;r)dr.$$ \Gamma_{k,\epsilon}(x,t;\rho)=\int_0^{\rho}g_{k,\epsilon}(x,t;r)dr. $$

We may invoke Gronwall’s lemma to get the estimate

∫Ω[∂∂tFε(x,t;Duε(t))+∂∂tΓk,ε(x,t;uε(t))]dx≤c.$$ \int_{\Omega}\left[\frac{\partial}{\partial t}F_{\epsilon}(x,t; Du_{\epsilon}(t))+\frac{\partial}{\partial t}\Gamma_{k,\epsilon}(x,t;u_{\epsilon}(t))\right]\, dx\leq c.$$

Therefore,

‖u˙ε‖L2(Q)≤c.$$ \|\dot{u}_{\epsilon}\|_{L^2(Q)}\leq c.$$

and consequently,

‖u˙ε‖2;2+‖uε‖2;2+‖uε‖p;m,p≤c,$$ \begin{equation} \|\dot{u}_{\epsilon}\|_{2;2}+\|u_{\epsilon}\|_{2;2}+\|u_{\epsilon}\|_{p;m,p}\leq c, \end{equation} $$(6)

where the constant c is independent of ε, k, and n.

From (6) and in view of Arzelà-Ascoli’s theorem, we get

uε→un strongly in Yn and u˙ε→u˙n(weakly) in L2(QT).$$ u_{\epsilon}\to u_n \text{strongly in} Y_n \text{and} \dot{u}_{\epsilon}\to\dot{u}_n \text{(weakly) in} L^2(Q_T).$$

Therefore, (5) yields

∫0T(u˙n,v−un)dt+∫0T(A(un),v−un)dt+∫QTgk(x,t,un)(v−un)dxdt≥∫QTf˜(v−un)dxdt,$$ \begin{equation} \int_0^T (\dot{u}_n,v-u_n)dt+\int_0^T(A(u_n),v-u_n)dt+\int_{Q_T}g_k(x,t,u_n)(v-u_n)dxdt\geq \int_{Q_T}\tilde{f}(v-u_n)\, dxdt, \end{equation} $$(7)

where v ∈ Y_n∩ K and u_n(0) = 0. By the lower-semicontinuity property of the norms in (6), we get (4).

From (4) and the fixes level of truncation, we get

{u˙n⇀u˙k (weakly) in L2(QT)un→uk strongly in Lp(0,T;W0m−1,p(Ω)) and weakly in C(0,T,L2(Ω))Aα(x,t;Dun)⇀hα(x,t) (weakly) in Lp′(QT),|α|≤mgk(x,t;un)→gk(x,t;uk) strongly in Lp′(QT).$$ \begin{equation} \begin{cases} \dot{u}_n\rightharpoonup \dot{u}_k \text{ (weakly) in } L^2(Q_T)\\ u_n\to u_k \text{ strongly in } L^p(0,T;W_0^{m-1,p}(\Omega)) \text{ and weakly in } C(0,T,L^2(\Omega))\\ A_{\alpha}(x,t;Du_n)\rightharpoonup h_{\alpha}(x,t) \text{ (weakly) in } L^{p'}(Q_T), |\alpha|\leq m\\ g_k(x,t;u_n)\to g_k(x,t;u_k) \text{ strongly in } L^{p'}(Q_T). \end{cases} \end{equation} $$(8)

Secondly, we show that u_k is a weak solution of (3). For this purpose and in view of (6), it suffices to prove that

limsupn∫0T(A(un),un−uk)dt≤0.$$ \begin{equation} \limsup_n \int_0^T (A(u_n),u_n-u_k)dt\leq 0. \end{equation} $$(9)

This inequality holds true at least for one subsequence (vk)⊂∪n=1∞Yn $(v_k)\subset \cup_{n=1}^{\infty}Y_n$ . From (7), for any fixed k, we have

∫0T(u˙n,vk−un)dt+∫0T(A(un),vk−un)dt+∫QTgk(x,t,un)(vk−un)dxdt≥∫QTf˜(vk−un)dxdt.$$ \begin{equation} \int_0^T (\dot{u}_n,v_k-u_n)dt+\int_0^T(A(u_n),v_k-u_n)dt+\int_{Q_T}g_k(x,t,u_n)(v_k-u_n)\, dxdt\geq \int_{Q_T}\tilde{f}(v_k-u_n)\, dxdt. \end{equation} $$(10)

Let v_k be the truncation at level k and the mollification with respect to the time and space variables, respectively, of the Galerkin’s solution u_n, i.e., vk=((unk)μ)σ $v_k=((u_n^k)_{\mu})_{\sigma}$ . Letting n → ∞ in (10), taking (8) into account, and the strong convergence of ((unk)μ)σ $((u_n^k)_{\mu})_{\sigma}$ into u_k in X with respect to σ, μ, [6], we obtain (9) and hence, u_k ∈ W ∩ K is a weak solution of (3), i.e.,

∫0T(u˙k,v−uk)dt+∫0T(A(uk),v−uk)dt+∫QTgk(x,t,uk)(v−uk)dxdt≥∫QTf˜(v−uk)dxdt$$ \begin{equation} \int_0^T (\dot{u}_k,v-u_k)dt+\int_0^T (A(u_k),v-u_k)dt+\int_{Q_T}g_k(x,t,u_k)(v-u_k)\, dxdt\geq \int_{Q_T}\tilde{f}(v-u_k)\, dxdt \end{equation} $$(11)

for all v ∈ W ∩ K. Finally, to show (2), it remains to prove the following assertions:

u˙k⇀u˙(weakly) inL2(QT),$$ \begin{equation} \dot{u}_k\rightharpoonup \dot{u} \text{(weakly) in} L^2(Q_T), \end{equation} $$(12)uk→u(strongly) inLp(0,T;W0m−<!-- - -->1,p(Ω)),$$ \begin{equation} u_k\to u \,\,\text{(strongly) in} L^p(0,T;W_0^{m-1,p}(\Omega)), \end{equation} $$(13)and (weakly) in C(0,T;L2(Ω)),$$ \begin{equation} \text{and (weakly) in } C(0,T;L^2(\Omega)), \end{equation} $$(14)gk(x,t;uk)→g(x,t;u)(strongly) in L1(QT),$$ \begin{equation} g_k(x,t;u_k)\to g(x,t;u) \text{(strongly) in } L^1(Q_T), \end{equation} $$(15)

and

Dαuk(x,t)→Dαu(x,t)a.e. in QT for all |α|≤m.$$ \begin{equation} D^{\alpha}u_k(x,t)\to D^{\alpha}u(x,t) \text{a.e. in } Q_T \text{for all} |\alpha|\leq m. \end{equation} $$(16)

Assertions (12-15) follow similarly as above and as in [5]. To show 16, it suffices to show

limsupk∫0T(A(un),uk−u)dt≤0.$$ \begin{equation} \limsup_k \int_0^T (A(u_n),u_k-u)dt\leq 0. \end{equation} $$(17)

Since for any v ∈ X we may find a sequence (v_ℓ) converging weakly to u, we get from (11)

∫0T(u˙k,uk)dt+∫0T(A(uk),uk−vℓ)dt+∫QTgk(x,t;uk)ukdxdt≤∫QTgk(x,t;uk)vℓdxdt+∫0T(u˙k,vℓ)dt−∫0T(f,vℓ−uk)dt.$$ \begin{align*} \int_0^T (\dot{u}_k,u_k)dt&+\int_0^T (A(u_k),u_k-v_{\ell})dt+\int_{Q_T} g_k(x,t;u_k)u_k\, dxdt\\ &\leq \int_{Q_T} g_k(x,t;u_k)v_{\ell}\, dxdt+\int_0^T (\dot{u}_k,v_{\ell})dt-\int_0^T (f,v_{\ell}-u_k)dt. \end{align*} $$

Letting k → ∞, keeping ℓ fixed, taking into account Fatou’s lemma, and then allowing ℓ → ∞, we obtain (17) and consequently, (2) follows.