数理解析研究所講究録
第 1892 巻 2014 年 73-87

73

A subdifferential approach to evolution equations
in variable exponent Lebesgue spaces
Goro Akagi

Graduate School of System Informatics,
Kobe University
Abstract

In this resum\’e, we review recent results reported in the paper [1],where a subdifferential approach to doubly nonlinear parabolic
equations involving variable exponents is proposed.

This note is based on a joint work with Giulio Schimperna (Pavia University,
Italy).

1

Introduction

Doubly nonlinear parabolic equations have been vigorously studied so far, as they appear in various fields such as phase transition, damage mechanics and fluid dynamics.
A typical example is as follows:$\beta(\partial_{t}u)-\Delta u=f$

in

(1)

$\Omega\cross(O, T)$with a maximal monotone graph :$T>0$ , and a given, a domain of$f=f(x,t):\Omega\cross(0,T)arrow\mathbb{R}$function. The linear Laplacian is often replaced witha nonlinear variant, e.g., the so-called $m$-Laplacian
given by$\beta$$\mathbb{R}arrow \mathbb{R}$$\Omega$$\mathbb{R}^{N},$$\triangle$$\triangle_{m}$$\triangle_{m}u=div(|\nabla u|^{m-2}\nabla u) , 1<m<\infty,$

and then, (1) is truly a doubly nonlinear parabolic equation. The doubly nonlinear
parabolic equation is also classified as a fully nonlinear equation, and due to thesevere nonlinearity, it is somewhat delicate which space is chosen as a base space, afunction space in which the equation is mainly treated throughout analysis, so as toapply an energy method in an effective way.
By setting $u(t):=u(\cdot, t)$ , such a nonlinear parabolic equation is interpreted as an
abstract evolution equation,$A(u’(t))+B(u(t))=f(t)$ in $X,$

$0<t<T$(2)

74
with unknown function $u:(0, T)arrow X$ , two (possibly nonhnear) operators $A,$ in aproper function space $X$ and : $(0, T)arrow X$ . Equation (2) is often called a doubly
nonlinear evolution equation.
In most of studies on nonlinear evolution equations, existence and regularity
results are usually established in a proper class of vector-valued functions, e.g., aLebesgue-Bochner space,
$B$

$f$

$L^{p}(0, T;X)$ $:=\{u$: $(0, T)arrow X$ : “strongly measurable” and$\int_{0}^{T}\Vert u(t)\Vert_{X}^{p}dt<\infty\}.$Here “strong measurability” of means that there exists a sequence of simple functions $u_{n}:(0, T)arrow X$ such that $u_{n}(t)arrow u(t)$ strongly in $X$ for a.e. $t\in(O, T)$ .
Now, let us recall several results on the existence and regularity of solutions forthe doubly nonlinear evolution equation (2) in Bochner space frameworks. Barbu [3],Arai [2] and Senba [12] obtained existence results based on the Hilbert space, $L^{2}(0, T;H)$ .
The method of their proofs relies on the time differentiation of the equation, whichtransforms the equation into another (more tractable) type of doubly nonlinear equations, and a peculiar monotonicity condition called an -monotonicity for , i.e.,
$u$

$B$

$A$

$(Bu-Bv, A_{\lambda}(u-v))_{H}\geq 0$

for all$u,$$v\in D(B)$and$\lambda>0,$

wheredenotes the Yosida approximation of . Colli-Visintin [7] and Colli [6]also treated (2) in the Hilbert space $L^{2}(0, T;H)$ and in the reflexive Banach space$L^{p}(0, T;V)$ with $1<p<\infty$ , respectively. However, their approach is totally different
from the former one, and instead of differentiating the equation and assuming the-monotonicity of , they impose a -power growth condition,$A$$A_{\lambda}$

$A$

$B$

$p$

$c_{\prec)}\Vert u\Vert_{V}^{p}\leq\langle \mathcal{A}u,$$u\rangle_{V}+C,$$\Vert Au\Vert_{V}^{p’}.$for $u\in V,$$\leq C(\Vert u\Vert_{V}^{p}+1)$

on the operator $A:Varrow V^{*}$ defined on a Banach space

and its dual space(inthe Hilbert space setting, $V=V^{*}=H$ ).
These results have been applied to various types of doubly nonhnear parabolic
equations, and here, let us take the following example:$V$

in $Q:=\Omega\cross(O, T)$ ,$|\partial_{t}u|^{p-2}\partial_{t}u-\Delta_{m}u=f(x, t)$

on

$u=0$

in

$u(\cdot, 0)=u_{0}$$\partial\Omega\cross(0, T)$$\Omega$$V^{*}$

(3)
(4)
(5)

,

,

where $1<p,$ $m<\infty$ . Equation (3) can be regarded as a sort of generalized GinzburgLandau equations proposed by Gurtin [9],$\beta(u, \nabla u, \partial_{t}u)\partial_{t}u=div[\partial_{\nabla u}\psi(u, \nabla u)]-\partial_{u}\psi(u, \nabla u)+\gamma$

with kinetic coefficient , free energy density and external microforce . In particular, (3) $-(5)$ is well suited to the general theory due to Colli [6] based on the Banach
space $L^{p}(0, T;V)$ by setting $V=IP(\Omega)$ , $Au=|u|^{p-2}u$ and $Bu=-A_{m}u$ . Indeed,$\beta$$\psi$$\gamma$

75
one can easily check that the -power growth condition holds true, more precisely, itholds that$\Vert u\Vert_{V}^{p}=\langle Au, u\rangle_{V}, \Vert Au\Vert_{V^{*}}^{p’}=\Vertu\Vert_{V}^{p}.$In this note, we shall treat a variant of (3) $-(5)$ involving variable exponents. Moreprecisely, letbe a smooth bounded domain and consider$\Omega\subset \mathbb{R}^{N}$$|\partial_{t}u|^{p(x)-2}\partial_{t}u-\Delta_{m(x)}u=f(x, t)$$u=0$

in $Q:=\Omega\cross(O, T)$ ,
,
on
in ,

(6)
(7)
(8)

$\partial\Omega\cross(0, T)$$u(\cdot, 0)=u_{0}$$\Omega$

where $1<p(x),$ $m(x)<\infty$ are variable exponents andLaplacian given by$\Delta_{m(x)}u=div(|\nabla u|^{m(x)-2}\nabla u)$$\Delta_{m(x)}$stands for the $m(x)-$.

It is worth mentioning that Equation (6) can describe mixed settings of several typesof (generalized) Ginzburg-Landau models, e.g., the case$\Omega=\Omega_{1}\oplus\Omega_{2}.$Not only does such a generalization extend the scope of the abstract theory developed
so far in order to cover Equation (6) but also it would shed new light on the theoryof evolution equations by reconsidering whether a vector-valued function space suchas a Lebesgue-Bochner space is an optimal choice as a base space.

2

Lebesgue and Sobolev spaces with variable exponents

In this section, we briefly review some material on variable exponent Lebesgue andSobolev spaces and set up notation. We refer the reader to [8] as a survey of thisfield.
Define the set of variable exponents by$\mathcal{P}(\Omega):=\{p\in \mathcal{M}(\Omega):eSxS\in|_{\iota}^{nfp(x)}\geq 1\},$

where

denotes the set of Lebesgue measurable functions defined on, denote the (essential) supremum and infimum of $p(x)$ by$\mathcal{M}(\Omega)$$p(x)\in \mathcal{P}(\Omega)$$p^{-}:= ess\inf_{x\in Jl}p(x)$and$p^{+}:= ess\sup_{x\in\Omega}p(x)$and define the class of $log$ -H\"older continuous variable exponents by$\mathcal{P}_{\log}(\Omega):=\{p\in \mathcal{P}(\Omega):|p(x)-p(x’)|\leq\frac{L}{\log(|x-x’|^{-1}+e)}$

for all$x,$$x’\in\Omega$and some $L>0$ .

$\Omega$

. For

76
Now, variable exponent Lebesgue and Sobolev spaces are defined by$L^{p(x)}( \Omega) :=\{u\in \mathcal{M}(\Omega):\int_{tl}|u(x)|^{p(x)}dx<\infty\}$with norm$\Vert u\Vert_{L^{p(x)}(\{\})} :=\inf\{\lambda>0:\int_{\{\}}|\frac{u(x)}{\lambda}|^{p(x)}dx\leq 1\},$

and$W^{1,p(x)}(\Omega)$$:=\{u\in L^{p(x)}(\Omega):\partial_{x_{i}}u\in L^{p(x)}(\Omega)$

for $i=1,$$\ldots,$$N\},$

whose norm is given by$\Vert u\Vert_{W^{1,p(x)}(\Omega)}:=(\Vert u\Vert_{L^{p(x)}(\})}^{2}+\Vert\nablau\Vert_{L^{p(x)}(1l)}^{2})^{1/2}$

3

Difficulties arising from variable exponents

This section is devoted to discussing difficulties of treating Equation (6),$|\partial_{t}u|^{p(x)-2}\partial_{t}u-\Delta_{m(x)}u=f(x, t)$,arising from the presence of variable exponents. Following a classical strategy as inconstant exponent cases, we set$V=L^{p(x)}( \Omega)=\{u\in \mathcal{M}(\Omega):\int_{1l}|u(x)|^{p(x)}dx<\infty\}$and note that$(|u|^{p(x)-2}u)u=|u|^{p(x)}, ||u|^{p(x)-2}u|^{p’(x)}=|u|^{p(x)}.$However, it only implies$\langle|u|^{p(x)-2}u, u\rangle_{V}\geq c\Vert u\Vert_{V}^{p^{-}}, \Vert|u|^{p(x)-2}u\Vert_{V^{*}}^{(p^{+})’}\leq C\Vert u\Vert_{V}^{p^{+}}$

with positive constants

$c,$

$C$

and $V^{*}=L^{p’(x)}(\Omega)$ . Since $p^{+}>p^{-}$ , the equation does

not fall within the scope of the general theory of [6].
Let us more precisely discuss a difficulty due to the relation $p^{+}>p^{-}$ pecuhar tothe variable exponent setting. For simplicity, suppose $f\equiv 0$ and test (6) bytosee that$\partial_{t}u$$\int_{l}|\partial_{t}u|^{p(x)}dx+\frac{d}{dt}\int_{tl}\frac{1}{m(x)}|\nabla u|^{m(x)}dx=0.$

The integration of both sides over $(0, t)$ implies$\int_{0}^{t}\int_{\Omega}|\partial_{\tau}u|^{p(x)}dxd\tau+\int_{\Omega}\frac{1}{m(x)}|\nabla u(x, t)|^{m(x)}dx\leq\int_{\Omega}\frac{1}{m(x)}|\nabla u_{0}(x)|^{m(x)}dx$

.

(9)

77
To estimate$|\partial_{t}u|^{p(x)-2}\partial_{t}u$

in $V^{*}=L^{p’(x)}(\Omega)$ , we use the relation,$\int_{1l}||\partial_{t}u|^{p(x)-2}\partial_{t}u|^{p’(x)}dx=\int_{tl}|\partial_{t}u|^{p(x)}dx$

.

(10)

If one works in a framework (called “Flrrame ” below) based on a Bochner space$(e.g., L^{p}(0, T;V))$ , one needs to derive estimates forandin a proper
Bochner space and its dual space, respectively. However, in the variable exponent
setting, we should pay attention to a gap between the modular and norm of,that is,$B$$\partial_{t}u$$|\partial_{t}u|^{p(x)-2}\partial_{t}u$$L^{p(x)}(\Omega)$$\int_{Il}|w(x)|^{p(x)}dx\neq\Vert w\Vert_{L^{p(x)}(11)}^{p(x)}$for $w\in L^{p(x)}(\Omega)$ .

To overcome this defect, we usually use the following relation between the modular
and norm:$\sigma_{p(x)}^{-}(\Vert w\Vert_{Lp(x)})\leq\int_{ll}|w(x)|^{p(x)}dx\leq\sigma_{p(x)}^{+}(\Vert w\Vert_{L^{p(x)}})$

with$\sigma_{p(x)}^{-}(s)$estimates in$:= \max\{s^{p}‘, s^{p^{+}}\}$ . Then one may obtainandchner spaces with some loss of integrability in (cf. see (9)) such as$:= \min\{s^{p^{-}}, s^{p^{+}}\}$$Bo$

for all $w\in L^{p(x)}(\Omega)$$\sigma_{p(x)}^{+}(s)$$t$$\int_{0}^{T}\Vert\partial_{t}u\Vert_{V}^{p^{-}}dt\leq\int_{0}^{T}(\int_{Il}|\partial_{t}u|^{p(x)}dx)dt,$$\int_{0}^{T}\Vert|\partial_{t}u|^{p(x)-2}\partial_{t}u\Vert_{V}^{(p}|^{)^{-}}dt\leq\int_{0}^{T}(\int_{tl}|\partial_{t}u|^{p(x)}dx)dt.$

However, there is no duality between two spaces $IP^{-}(0, T;V)$ and $L^{(p’)^{-}}(0, T;V^{*})$ ,
where
and
, respectively, are estimated. Indeed, we find that$L^{(p’)^{-}}(0, T;V^{*})$
is not identified with the dual space of $L^{p^{-}}(0, T;V)$ due to the factthat $(p’)^{-}=(p^{+})’<(p^{-})’$ by $p^{+}>p^{-}$ Such a lack of a duality prevents us to identify
the limit of approximate solutions in a usual manner so as to prove the existence ofsolutions.
On the other hand, a framework (called “Frame ” below) based on Lebesgue
spaces for functions of space-time variables shows us a different picture. Recall the
relation (10). Then we immediately observe that$\partial_{t}u$$|\partial_{t}u|^{p(x)-2}\partial_{t}u$$L$$\iint_{Q}1\partial_{t}u|^{p(x)-2}\partial_{t}u|^{p’(x)}dxdt=\iint_{Q}|\partial_{t}u|^{p(x)}dxdt$

with $Q:=\Omega\cross(0, T)$ , which implies, e.g.,$\Vert|\partial_{t}u|^{p(x)-2}\partial_{t}u\Vert_{L^{p’(x)}(Q)}\leq(\Vert\partial_{t}u\Vert_{L^{p(x)}(Q)})^{p^{+}/p^{-}}$(see also (9)). Here it is noteworthy that there is no loss of integrabihty through theseprocedures. Moreover, since $L^{p’(x)}(Q)$ is identified with the dual space of $L^{p(x)}(Q)$ ,there is a duahty between two spaces in which the estimate is established. Hence the78
operator
is well defined from $L^{p(x)}(Q)$ into its dual space
Moreover, we observe that$\mathcal{A}:u\mapsto|u|^{p(x)-2}u$$\mathcal{A}:L^{p(x)}(Q)arrow L^{p’(x)}(Q)$$I\nearrow’(x)(Q)$.

is bounded and coercive.

Therefore it seems better to work in the Lebesgue space, $L^{p(x)}(Q)$ , in order totreat Equation (6) without any loss of integrability (in t) (cf. we found loss in view ofFrame B). On the other hand, in contrast with constant exponent cases, there is no
Bochner space which can be identified with the Lebesgue space $If^{(x)}(Q)$ , as the formal
description $L^{p(x)}(0, T;IP^{(x)}(\Omega))$ has no longer sense due to the -dependence of $p(x)$ .
Furthermore, in most of studies on evolution equations in view of energy methods, thechain-rule for gradient operators (e.g., subdifferential) is often employed and plays
an crucial role. However, chain-rules are always formulated and proved in Frame(see, e.g., [5] and [10]). So this situation encourages us to develop a combination of
two frameworks, EYame and Frame , in a suitable way.
$x$

$B$

$B$

4

$L$

Main results of [1]The main results of [1] are concerned with the existence and regularity of solutions
for the Cauchy-Dirichlet problem (6) $-(8)$ . To prove these results, we shall present amixed framework of Frame and Frame L. Moreover, we shall develop some devicesof subdifferential calculus, in particular, a chain-rule for subdifferentials in the mixed
frame.
To state the main results, let us introduce basic assumptions (H),$B$

$m\in \mathcal{P}_{\log}(\Omega) , p\in \mathcal{P}(\Omega) , 1<p^{-}, m^{-},p^{+}, m^{+}<\infty$

,

(Hl)

$ess\inf_{x\in l}(m^{*}(x)-p(x))>0, m^{*}(x):=\frac{Nm(x)}{(N-m(x))_{+}}$

,

(H2)

$f\in L^{p’(x)}(Q) , u_{0}\in W_{0}^{1,m(x)}(\Omega)$

.

(H3)

Remark 4.1. (i) By (Hl),arable Banach spaces.
(ii) Since$m(\cdot)\in \mathcal{P}_{\log}(\Omega)$$L^{p(x)}(\Omega)$and, one can define$W^{1,m(x)}(\Omega)$$W_{0}^{1,p(x)}(\Omega)$

are uniformly convex and sep-by

$W_{0}^{1,m(x)}(\Omega):=\overline{C_{0}^{\infty}(\Omega)}^{W^{1,m(x)}(t\})}, \Vert u\Vert_{W_{0}^{1,m(x)}(\Omega)}:=\Vert\nabla u\Vert_{L^{m(x)}(\Omega)},$and moreover, it has similar properties (e.g., Poincar\’e and Sobolev inequalities)
to the constant exponent case.
(iii) Moreover, (H2) ensures that

$W_{0}^{1,m(x)}(\Omega)^{com}\hookrightarrow^{pact}IP^{(x)}(\Omega)$.

We are concerned with strong solutions of (6) $-(8)$ defined by

79
Definition 1 (Strong solutions)
We call $u\in U^{(x)}(Q)$ a strong solution of (6) $-(8)$ in
conditions hold true:
(i)

(ii)

is continuous with values inand it is weakly continuous with values in$t\mapsto u(\cdot, t)$$\partial_{t}u\in L^{p(x)}(Q),$$L^{p(x)}(\Omega)$$\triangle_{m(x)}u\in L^{p’(x)}(Q)$$Q$

whenever the following

on $[0, T],$
on $[0, T],$

$W_{0}^{1,m(x)}(\Omega)$,(iii) the equation (6) holds for a.e. $(x, t)\in Q,$(iv) the initial condition (8) is satisfied for a.e.

$x\in\Omega.$

In [1], the following theorems are proved.
Theorem 2 (Existence of strong solutions [1])Assume (H). Then the Cauchy-Dirichlet problem (6) $-(8)$ admits (at least) one
strong solution$u.$Theorem 3 (Time-regularization of strong solutions [1])In addition to (H), suppose that$t\partial_{t}f\in L^{p’(x)}(Q)$.

Then, the Cauchy-Dirichlet problem (6) $-(8)$ admits a strong solution
additionally satisfies$u$, which$ess\sup_{t\in(\delta,T)}\Vert\partial_{t}u(\cdot, t)\Vert_{L^{p(x)}(tl)}<\infty,$$ess\sup_{t\in(\delta,T)}\Vert\triangle_{m(x)}u(\cdot, t)\Vert_{L^{p’(x)}(tl)}<\infty$

for any $\delta\in(0, T)$ .

5

Two formulations of the equation

We first set up a formulation based on a Bochner space setting, “Frame $B,$ ” for(6) $-(8)$ . Set$V=L^{p(x)}(\Omega)$
and $X=W_{0}^{1,m(x)}(\Omega)$with norms$\Vert u\Vert_{V}:=\Vert u\Vert_{L^{p(x)}(tl)},$$\langle v,$$\Vert u\Vert_{X}:=\Vert\nabla u\Vert_{L^{m(x)}(t1)}$$u \rangle_{V}=\int_{tl}u(x)v(x)dx$

By (H2), it follows that $X$

compact$V$and duality pairing

for all $u\in V,$ $v\in V^{*}=L^{p’(x)}(\Omega)$ .
andcompact.

$V^{*}\hookrightarrow X^{*}$

80
Define functionals$\psi$

andon$\phi$$V$by
for $u\in V$$\psi(u):=\int_{\zeta\}}\frac{1}{p(x)}|u(x)|^{p(x)}dx$$\phi(u):=\{\begin{array}{ll}\int_{\Omega}\frac{1}{m(x)}|\nabla u(x)|^{m(x)}dx if u\in X,+\infty if u\in V\backslash X.\end{array}$Denote by$\partial_{\zeta\}}$the subdifferential in $V=L^{p(x)}(\Omega)$ . Then (6) $-(8)$ can be reduced to$\partial_{\Omega}\psi(u’(t))+\partial_{\Omega}\phi(u(t))=Pf(t)$in$0<t<T,$$V^{*},$$u(O)=u_{0},$where $Pf(t)$ $:=f(\cdot, t)$ . Here we emphasize that the notion of a subdifferential isessentially needed here. Indeed, we work in $V=If^{(x)}(\Omega)$ to get rid of any loss of
integrabihty in ; however, the functional is not smooth in . So a notion of thederivative for non-smooth functionals is required.
Next, we transform the formulation in “FYame ” into one in “Frame L.” To thisend, we carefully reconsider the correspondence between functions in two frameworks, writeby taking account of variable exponent Lebesgue spaces. For each
$x$

$V$

$\phi$$B$$u\in \mathcal{M}(Q)$

Pu $(t):=u(\cdot, t)$

for $t\in(O, T)$ .

Then it follows that
Proposition 4 (Identification between $B-$ and -spaces [1])$L$

Let $1\leq p<\infty$ and let $p(x)$ be such that $1\leq p^{-}\leq p^{+}<\infty.$(i) $P$ is a linear, bijective, isometric mapping from $L^{p}(Q)$ to $L^{p}(0, T;L^{p}(\Omega))$ .
Furthermore, if $u\in L^{p(x)}(Q)$ , then $Pu\in\nu^{-}(0, T;U^{(x)}(\Omega))$ .

(ii) The inverse$P^{-1}$: $L^{p}(0, T;L^{p}(\Omega))arrow L^{p}(Q)$ is well defined, and for $u=$
for a.e. $t\in(O, T)$ .

$u(t)\in L^{p}(0, T;L^{p}(\Omega)),$ $u(t)=P^{-1}u(\cdot, t)$(iii) If $u\in L^{p(x)}(Q)$ with $\partial_{t}u\in L^{p(x)}(Q)$ , then $Pu$ belongs to the space$W^{1,p^{-}}(0, T;L^{p(x)}(\Omega))$ and (Pu)’.
$=P(\partial_{t}u)$(iv) If $u\in W^{1,p}(0, T;L^{p}(\Omega))$ , thenRemark 5.1. It is known that$\partial_{t}(P^{-1}u)\in L^{p}(Q)$$L^{\infty}(0, T;L^{\infty}(\Omega))$and $\partial_{t}(P^{-1}u)=P^{-1}(u’)$ .

is not identified with$L^{\infty}(Q)$(see [11]).

Set$\mathcal{V}:=L^{p(x)}(Q)$and$\mathcal{V}^{*};=L^{p’(x)}(Q)$with $Q=\Omega\cross(0, T)$ .

Let : $V(=\nu^{(x)}(\Omega))arrow(-\infty, \infty]$ be a proper lower semicontinuous convex funcby
tional and define :$\varphi$$\Phi$$\mathcal{V}arrow(-\infty, \infty]$$\Phi(u);=\{\begin{array}{ll}\int_{0}^{T}\varphi(Pu(t))dt if \varphi(Pu(\cdot))\in L^{1}(0, T) ,\infty otherwise.\end{array}$81
Here and henceforth, denote by
the subdifferential inthat
Proposition 5 (Identification of subdifferentials [1])$\mathcal{V}=L^{p(x)}(Q)$$\partial_{Q}$For$u\in \mathcal{V},$$\xi\in \mathcal{V}^{*}$. Then it holdswith $1<p^{-}\leq p^{+}<\infty,$iff$\xi\in\partial_{Q}\Phi(u)$$P\xi(t)\in\partial_{tl}\varphi(Pu(t))$

for a.e. $t\in(O, T)$ .

Now, we are ready to provide a formulation of (6) $-(8)$ based on Frame L. Definefunctionals and on by$\Psi$$\Phi$$\mathcal{V}$$\Psi(u) :=\iint_{Q}\frac{1}{p(x)}|u(x, t)|^{p(x)}dxdt=\int_{0}^{T}\psi(Pu(t))dt,$$\Phi(u):=\{\begin{array}{l}\int_{0}^{T}\phi(Pu(t))dt if Pu (t)\in X for a.e.

for$u\in \mathcal{V}$t\in(O, T) ,t\mapsto\phi(Pu(t))\in L^{1}(0, T) ,\infty otherwise\end{array}$. Then by Proposition 5, the evolution equation$\partial_{\Omega}\psi(u’(t))+\partial_{\Omega}\phi(u(t))=Pf(t)$in$(\Leftrightarrow(6)-(8))$,$0<t<T,$$V^{*},$

is equivalently rewritten as the relation,$\partial_{Q}\Psi(\partial_{t}(P^{-1}u))+\partial_{Q}\Phi(P^{-1}u)=f$6

in$\mathcal{V}^{*}$

Construction of a strong solution

In this section, we give an outline of a proof for Theorem 2.

Step 1 (Time-discretization) We consider the following time-discretized equations, for $n=0,$$N-1,$$\ldots,$$\partial_{\Omega}\psi(\frac{u_{n+1}-u_{n}}{h})+\partial_{\Omega}\phi(u_{n+1})=f_{n+1}$

in$V^{*}$

with$h:=T/N,$$t_{n}:=nh$and$f_{n}:= \frac{1}{h}\int_{t_{n-1}}^{t_{n}}Pf(\theta)d\theta.$

The existence of $u_{n+1}\in X$ can be proved by using a variational method.
Moreover, define a piecewise forward constant interpolant: $(0, T)arrow X=$
and a piecewise linear interpolant: $(0, T)arrow X$ by$\overline{u}_{N}$$W_{0}^{1,m(x)}(\Omega)$$\overline{u}_{N}(t):=u_{n+1},$$u_{N}$$u_{N}(t);= \frac{t-t_{n}}{h}u_{n+1}+\frac{t_{n+1}-t}{h}u_{n}$for $t\in[t_{n}, t_{n+1})$ .

82Then we have$\partial_{tl}\psi(u_{N}’(t))+\partial_{\zeta\}}\phi(\overline{u}_{N}(t))=\overline{f}_{N}(t)$with $u_{N}(0)=u_{0}$ in Frame

$B$

in

$V^{*}$

for a.e.

,$t\in(O,T)$, and equivalently,
in$\partial_{Q}\Psi(\partial_{t}(P^{-1}u_{N}))+\partial_{Q}\Phi(P^{-1}\overline{u}_{N})=P^{-1}\overline{f}_{N}$$\mathcal{V}^{*},$$u_{N}(0)=u_{0}$in Frame L.
Step 2 (Energy estimates) Test the discretized equation by $(u_{n+1}-u_{n})/h$ toobtain$\iint_{Q}|\partial_{t}(P^{-1}u_{N})|^{p(x)}dxdt+\sup_{t\in[0,T]}\phi(\overline{u}_{N}(t))\leq C,$

which also gives$\Vert\partial_{t}(P^{-1}u_{N})\Vert_{\mathcal{V}}\leq C, \sup|\overline{u}_{N}(t)|_{X}+\sup|u_{N}(t)|_{X}\leq C.$$t\in[0,T] t\in[0,T]$

Recall that$\mathcal{A}=\partial_{Q}\Psi$: $v\mapsto|v|^{p(x)-2}v$ is bounded from$\mathcal{V}$to$\mathcal{V}^{*}$

Thus we conclude that$\Vert\partial_{Q}\Psi(\partial_{t}(P^{-1}u_{N}))\Vert_{\mathcal{V}^{*}}\leq C,$

which also implies the boundedness of$\partial_{Q}\Phi(P^{-1}\overline{u}_{N})$in$\mathcal{V}^{*}$

by comparison.

Step 3 (Convergence) Passing to the limit as $Narrow\infty$ , up to subsequence, one
has the following convergences in both frames:$u_{N}arrow u$$\overline{u}_{N}arrow u$$P^{-1}\overline{u}_{N}arrow\hat{u}=P^{-1}u$$\partial_{t}(P^{-1}u_{N})arrow\partial_{t}\hat{u}$$\partial_{Q}\Phi(P^{-1}\overline{u}_{N})arrow\xi$$\partial_{Q}\Psi(\partial_{t}(P^{-1}u_{N}))arrow\eta$strongly in $C([O, T];V)$ ,weakly star in $L^{\infty}(0, T;X)$ ,strongly in $L^{\infty}(O, T;V)$ ,weakly star in $L^{\infty}(0, T;X)$ ,strongly in
weakly in
weakly in
weakly in$\mathcal{V},$$\mathcal{V},$$\mathcal{V}^{*},$$\mathcal{V}^{*}$, one caninThus $\eta+\xi=f$ in . From the maximal monotonicity of.
immediately obtain. To this end, we shall use Minty’s trick.
Now, it remains to show
One observes that$\mathcal{V}^{*}$$\partial_{Q}\Phi$$\mathcal{V}\cross \mathcal{V}^{*}$$\xi\in\partial_{Q}\Phi(\hat{u})$$\eta\in\partial_{Q}\Psi(\partial_{t}\hat{u})$$\int\int_{Q}\partial_{Q}\Psi(\partial_{t}(P^{-1}u_{N}))\partial_{t}(P^{-1}u_{N})dxdt$$= \int\int_{Q}(P^{-1}\overline{f}_{N}-\partial_{Q}\Phi(P^{-1}\overline{u}_{N}))\partial_{t}(P^{-1}u_{N})dxdt$$\leq\int\int_{Q}(P^{-1}\overline{f}_{N})\partial_{t}(P^{-1}u_{N})dxdt-\phi(u_{N}(T))+\phi(u_{0})$.

83Passing to the hmit as $Narrow\infty$ , we have$\lim_{narrow}\sup_{\infty}\int\int_{Q}\partial_{Q}\Psi(\partial_{t}(P^{-1}u_{N}))\partial_{t}(P^{-1}u_{N})dxdt$$\leq\int\int_{Q}f\partial_{t}\hat{u}dxdt-\phi(u(T))+\phi(u_{0})=?\int\int_{Q}\eta\partial_{t}\hat{u}dxdt.$

However, the last equality is not obvious at this moment, due to the lack of a chainrule for the current situation. To justify the equality, we need a new chain-rule forsubdifferential operators in a mixed framework.
Proposition 6 (Chain rule in a mixed frame [1])Letsatisfy $1<p^{-}\leq p^{+}<\infty$ . Letbe such that
Suppose that there exists
such that. Then, the function$t\mapsto\varphi(Pu(t))$ is absolutely continuous$[0,T]$. Moreover, for each $t\in(0, T)$ ,over$p(\cdot)\in \mathcal{P}(\Omega)$$u\in \mathcal{V}$$\partial_{t}u\in \mathcal{V}.$$\xi\in\partial_{Q}\Phi(u)$$\xi\in \mathcal{V}^{*}$

we have$\frac{d}{dt}\varphi(Pu(t))=\langle\eta,$

whenever $Pu$ and, we have$\varphi(Pu(\cdot))$$(Pu)’(t)\rangle_{V}$for all$\eta\in\partial_{Jl}\varphi(Pu(t))$,

are differentiable at . In particular, for $0\leq s<t\leq$
$t$

$T$

$\varphi(Pu(t))-\varphi(Pu(s))=\int\int_{\Omega\cross(s,t)}\xi\partial_{\tau}udxd\tau.$

Applying Proposition 6, we deduce that$\lim_{narrow}\sup_{\infty}\int\int_{Q}\partial_{Q}\Psi(\partial_{t}(P^{-1}u_{N}))\partial_{t}(P^{-1}u_{N})dxdt$$\leq \int\int_{Q}f\partial_{t}\hat{u}dxdt-\phi(u(T))+\phi(u_{0})$$u=P \hat{u}=\iint_{Q}f\partial_{t}\hat{u}dxdt-\phi(P\hat{u} (T))$$+\phi$( $P$ \^u( ))$O$$Prop6=\iint_{Q}f\partial_{t}\hat{u}dxdt-\iint_{Q}\xi\partial_{t}\hat{u}dxdt^{\xi+\eta=f}=\iint_{Q}\eta\partial_{t}\hat{u}dxdt,$

whence follows$\eta\in\partial_{Q}\Psi(\partial_{t}\hat{u})$Consequently,

7$\hat{u}$.

solves (6) $-(8)$ .

$\square$Outline of a proof for Proposition 6

In the section, we exhibit an outline of a proof for the new chain-rule.
Step 1 (Modification of the Moreau-Yosida approximation) We start withapproximating the proper lower semicontinuous convex functional defined on $V=$$\varphi$84$IP^{(x)}(\Omega)$by$:= \min_{v\in V}(\int_{t1}\frac{\lambda}{p(x)}|\frac{v(x)-u(x)}{\lambda}|^{p(x)}dx+\varphi(v))$$\varphi_{\lambda}(u)$for $u\in V,$

enjoys similar
which is called a modified Moreau-Yosida regularization of . Then
properties to the usual Moreau-Yosida regularization with the modified resolventofdefined below.
and modified Yosida approximation
Let $A:Varrow V^{*}$ be a maximal monotone operator.
$\varphi_{\lambda}$$\varphi$$J_{\lambda}$$A=\partial_{\{\}}\varphi$$A_{\lambda}$$\bullet$: $Varrow V$ of
The modified resolvent
which is a unique solution of$J_{\lambda}$$A$is given by, for each $u\in V,$in$Z_{(\}}( \frac{u_{\lambda}-u}{\lambda})+A(u_{\lambda})\ni 0$$J_{\lambda}u:=u_{\lambda},$$V^{*},$

where $Z_{tl}(u);=|u|^{p(x)-2}u$ for $u\in V.$$\bullet$

The modified Yosida approximation$A_{\lambda}(u)$on$A_{\lambda}$: $Varrow V^{*}$ of$A$

is given by

for each $u\in V.$$:=Z_{1}( \frac{u-J_{\lambda}u}{\lambda})\in A(J_{\lambda}u)$

One can also define the modified Moreau-Yosida regularization$\Phi_{\lambda}$of$\Phi$

defined$\mathcal{V}.$) Now, we have the following corresponStep 2 (Correspondence ofand:dence between
and)Lemma 7 (Correspondence ofand$\Phi_{\lambda}$$\varphi_{\lambda}$$\Phi_{\lambda}$$\varphi_{\lambda}$$\varphi_{\lambda}$$\Phi_{\lambda}[1]$It follows that$\Phi_{\lambda}(u)=\int_{0}^{T}\varphi_{\lambda}(Pu(t))dt$

In particular, for$u\in \mathcal{V}$$\xi_{\lambda}=\partial_{Q}\Phi_{\lambda}(u)$

and
iff

for all$u\in \mathcal{V}.$$\xi\in \mathcal{V}^{*},$

for a.a. $t\in(O, T)$ .

$P\xi_{\lambda}(t)=\partial_{tl}\varphi_{\lambda}(Pu(t))$

A similar relation is known for a setting based on Hilbert spaces $H$ and$L^{2}(0, T;H)$ .
However, it cannot be directly extended to Banach spaces$:=U(0, T;V)$ for $p\neq 2.$$\mathcal{H}$

$V$

$;=$

and

$\mathcal{V}$

Step 3 (Chain-rule for ) Thanks to the notion of the modified Moreau-Yosida
regularization, one shall obtain higher integrability for the subdifferentials of reguand be able to apply a standard chain-rule tolarized functionals, we deduce that
. Then sinceLetbe such that$Pu\in W^{1,p^{-}}(0, T;\nu^{(x)}(\Omega))$ (see Proposition 4). Moreover, since
is bounded, we
see that$\varphi_{\lambda}$$\varphi_{\lambda}.$$\varphi_{\lambda}$$u\in \mathcal{V}$$\partial_{t}u\in \mathcal{V}$$u,$ $\partial_{t}u\in\nu^{(x)}(Q)$$\partial\varphi_{\lambda}$$\partial_{\zeta\}}\varphi_{\lambda}(Pu(\cdot))\in L^{\infty}(0, T;V)$

.

85
Let

$\xi_{\lambda}$$:=\partial_{Q}\Phi_{\lambda}(u)$and use a standard chain-rule in Frame$\varphi_{\lambda}(Pu(t))-\varphi_{\lambda}(Pu$ $(s))^{ch}=^{ain}l^{t}\langle\partial_{Il}\varphi_{\lambda}$( $Pu$$(\tau)$),$B$

to obtain$(Pu)’(\tau)\rangle_{V}d\tau$$Lem7=\int_{s}^{t}\langle P\xi_{\lambda}(\tau), (Pu)’(\tau)\rangle_{V}d\tau$$= \int\int_{\Omega\cross(s,t)}\xi_{\lambda}\partial_{t}udxd\tau, 0\leq s\leq t\leq T.$

Step 4 (Convergence) To discuss the convergence of both sides of the relation as, we first note that
Lemma 8 (Boundedness of modified Yosida approx. [1])$\lambdaarrow 0$

Let $u\in V,$ $\eta\in Au$ and let
follows that

$A_{\lambda}$

be the modified Yosida approximation. Then it$\int_{\Omega}\frac{1}{p’(x)}|A_{\lambda}u(x)|^{p’(x)}dx\leq\int_{\Omega}\frac{1}{p’(x)}|\eta(x)|^{p’(x)}dx.$$\mathcal{V}^{*}An$.

analogous statement also holds for any maximal monotone operator

Thus since$\xi_{\lambda}=\partial_{Q}\Phi_{\lambda}$(Pu), we have, for any$\eta\in\partial_{Q}\Phi(Pu)$$\mathcal{A}:\mathcal{V}arrow$,$\int\int_{Q}\frac{1}{p’(x)}|\xi_{\lambda}|^{p’(x)}dxdt\leq\int\int_{Q}\frac{1}{p’(x)}|\eta|^{p’(x)}dxdt<\infty.$Hence$\xi_{\lambda}arrow\xi$

weakly in$\mathcal{V}^{*}$

and$\xi\in\partial_{Q}\Phi(Pu)$.

Thus$\varphi_{\lambda}(Pu(t))-\varphi_{\lambda}(Pu(s))=\int\int_{tlx(s,t)}\xi_{\lambda}\partial_{t}udxd\tau.$

Using the fact$\varphi_{\lambda}(u)arrow\varphi(u)$for all $u\in V,$

we have obtained the formula,$\varphi(Pu(t))-\varphi(Pu(\dot{s}))=\int\int_{Jl\cross(s,t)}\xi\partial_{t}udxd\tau,$

which also implies the absolute continuity of $t\mapsto\varphi(Pu(t))$ .

8$\square$Summary

In this note, we reviewed the results obtained in the paper [1]. The main results
are concerned with the existence and regularity (in time) of solutions of the CauchyDirichlet problem for the doubly nonlinear parabolic equation involving variable exponents,
in $Q:=\Omega\cross(O, T)$ .
$|\partial_{t}u|^{p(x)-2}\partial_{t}u-\Delta_{m(x)}u=f(x, t)$

86
Furthermore, we gave an outline of a proof for the existence result.
$\bullet$In order to efficiently use energy structures (without loss of integrabihty in ),we partially worked in “Frame ”, a framework based on the Lebesgue space
$t$

$L$

$\mathcal{V}:=L^{p(x)}(Q)$$\bullet$$\bullet$with $Q=\Omega\cross(0, T)$ .

To this end, we reformulated the problem both in “Rame ” and Fkame ”, aframework based on $Bo$ chner spaces, and also investigated the correspondences
between these frameworks.
$L$

$B$

We presented a new chain-rule for subdifferentials in a mixed framework. In its
statement, the assumptions are formulated in “Ftame ” and the conclusion isstated in the both frames.
$L$

References[1] Akagi, G. and Schimperna, G., Subdifferential calculus and doubly nonlinear evolution equations in $IP$ -spaces with variable exponents, submitted andarXiv: 1307. 2794 [math. AP].
[2] Arai, T., On the existence of the solution for $\partial\varphi(u’(t))+\partial\psi(u(t))\ni f(t)$ ,J. Fae. Sci. Univ. Tokyo Sec. IA Math. 26 (1979), 75-96.

[3] Barbu, V., Existence theorems for a class of two point boundary problems,
J. Diff. Eq. 17 (1975), 236-257.
[4] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach spaces,
Noordhoff, Leiden, 1976.

[5] Br\’ezis, H., Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol.5 North-Holland, Amsterdam/New York, 1973.
[6] Colli, P., On some doubly nonlinear evolution equations in Banach spaces, Japan
J. Indust. Appl. Math. 9 (1992), 181-203.
[7] Colli, P. and Visintin, A., On a class of doubly nonlinear evolution equations,Comm. Partial Differential Equations 15 (1990), 737-756.
, M., Lebesgue and Sobolev[8] Diening, L., Harjulehto, P., Haet\"o, P. andspaces with variable exponents, Lecture Notes in Mathematics, vol. 2017,
Springer-Verlag, Berhn, 2011.
$R\dot{u}\check{z}i\check{c}ka$[9] Gurtin, M.E., Generalized Ginzburg-Landau and Cahn-Hilliard equations basedon a microforce balance, Physica D92 (1996), 178-192.

87[10] Kenmochi, N., Some nonlinear parabolic variational inequalities, Israel J. Math.
22 (1975), 304-331.
[11] Roub\’i\v{c}ek, T., Nonlinear partial differential equations with applications, International Series of Numerical Mathematics, 153. Birkh\"auser Verlag, Basel, 2005.
[12] Senba, T., On some nonlinear evolution equation, Funkcial Ekvac. 29 (1986),

243-257.

Graduate School of System Informatics
Kobe University
1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501
JAPAN-mail address: akagi@port.kobe-u.ac.jp$E$$ffi\overline{\supset}\star\approx\neq\star\mp\mapsto^{\backslash }\beta_{\pi}^{A}$. シス$\overline{\mathcal{T}}A^{I}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\approx^{\backslash }\neq\Re_{Ju}^{*}7\backslash J\backslash$$*\prime\backslash *\mathbb{H}J\ovalbox{\tt\small REJECT}$