Doubly nonlinear evolution equations with non-monotone perturbations

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Doubly nonlinear evolution equations with non-monotone perturbations

Goro Akagi^∗1

1 Department of Machinery and Control Systems, School of Systems Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 330-8570, Japan.

The local (in time) existence of strong solutions to Cauchy problems for doubly nonlinear abstract evolution equations with non-monotone perturbations in reflexive Banach spaces is proved under appropriate assumptions, which allow the case where solutions of the corresponding unperturbed problem may not be unique. To prove the existence, a couple of approximate problems are introduced and delicate limiting procedures are discussed by using various tools from convex analysis and the Kakutani-Ky Fan fixed point theorem. Furthermore, an application of the preceding abstract theory to a nonlinear PDE is also given.

1 Introduction

LetV andV^∗be a reflexive Banach space and its dual space, respectively, and letH be a Hilbert space whose dual spaceH^∗ is identified with itself such that

V →H ≡H^∗→V^∗ (1)

with continuous and densely defined canonical injections. Letϕandψ^tbe proper lower semi-continuous functions fromV into(−∞,∞], and let∂_Vϕ, ∂_Vψ^t:V →2^V^∗ be subdifferential operators ofϕandψ^t, respectively, for eacht∈[0, T]. This talk deals with the existence of strong solutions for the following Cauchy problem:

(CP)

∂_Vψ^t(u(t)) +∂_Vϕ(u(t)) +B(t, u(t))f(t) inV^∗, 0< t < T, u(0) =u₀,

whereBdenotes an operator from(0, T)×V into2^V^∗ andf : (0, T)→V^∗andu₀ ∈D(ϕ) :={u∈V;ϕ(u)<∞}are given data.

For the unperturbed problem (B≡0), Colli [4] provided sufficient conditions for the existence of strong solutions to (CP) withψ^t≡ψ, and moreover, his results were extended to non-autonomous cases in [2]. As for perturbation problems (B≡0), Aso, Fr´emond and Kenmochi [3] proved the existence of time-global strong solutions for (CP) with∂_Vψ^t(u(t))replaced byA(u(t), u(t)), whereAis an operator fromH×H → 2^H, in the Hilbert space setting, i.e.,V = V^∗ = H. ˆOtani [5]

established an abstract theory on the existence of strong solutions for (CP) with∂_Vψ^t(u(t))replaced byu(t)in the Hilbert space setting. His results were applied to various nonlinear PDEs (e.g., quasilinear reaction-diffusion equation, Navier-Stokes equation).

In this study, we attempt to prove the existence of time-local strong solutions for (CP) by imposing appropriate conditions (the coerciveness, the boundedness and thet-smoothness of∂_Vψ^t, the precompactness of sub-level sets ofϕand the boundedness, the compactness and the measurability ofB) on the non-monotone operatorBas well as the functionalsϕ, ψ^t. We also emphasize that our abstract result is established in reflexive Banach space setting, and it covers the case where solutions may blow up in finite time.

As a typical example of nonlinear PDEs which fall within our abstract theory, we deal with the following initial-boundary value problem (IBVP):

|ut|^p−2u_t(x, t)−div(|∇u|^m−2∇u)(x, t)− |u|^q−2u(x, t) =f(x, t), (x, t)∈Ω×(0, T), u(x, t) = 0, (x, t)∈∂Ω×(0, T), u(x,0) =u₀(x), x∈Ω,

whereΩis a bounded domain inR^N,1< p, m, q <∞, andf : Ω×(0, T)→R,u₀: Ω→Rare given.

2 Main result

Before describing our main result, we introduce assumptions onψ^t, ϕandB. Letp∈(1,∞)andT >0be fixed.

∗ Corresponding author: e-mail:[email protected], Phone: +81 48 683 2020 Fax: +81 48 687 5197 PAMM · Proc. Appl. Math. Mech.7, 2040047–2040048 (2007) /DOI10.1002/pamm.200700647

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(A1) There exist positive constantsC_i(i= 1,2,3,4)such that C₁|u|^p_V ≤ψ^t(u) +C₂ for allt∈[0, T]andu∈D(ψ^t),

|η|^p_V∗ ≤C₃ψ^t(u) +C₄ for allt∈[0, T]and[u, η]∈∂_Vψ^t.

(A2) There exist a constantδ >0such that for allt₀∈[0, T]andv₀∈D(ψ^t⁰), we can take a function u:I_δ(t₀) := [t₀−δ, t₀+δ]→V satisfying

|u(t)−v₀|_V ≤ |α(t)−α(t₀)| ₀(|ψ^t⁰(v₀)|+|v₀|_V),

ψ^t(u(t))≤ψ^t⁰(v₀) +|β(t)−β(t₀)|) ₀(|ψ^t⁰(v₀)|+|v₀|_V)for allt∈I_δ(t₀) withα, β∈C([0, T])and a non-decreasing function ₀inR.

(A3) There exist a Banach spaceXand a non-decreasing function ₁inRsuch thatXis compactly embedded inV and|u|X≤ 1([ϕ(u)]++|u|H)for allu∈D(∂_Vϕ), where[s]+:= max{s,0}.

(A4) D(∂_Vϕ)⊂D(B(t,·))for a.e.t∈(0, T).For allε >0, there exist a constantC_ε≥0and a non-decreasing function ₂inRindependent ofεsuch that

|g|^p_V∗ ≤ε|ξ|^σ_V^∗+C_ε ₂(ϕ(u) +|u|V), whereσ:= min{2, p}, for a.e.t∈(0, T)and allu∈D(∂_Vϕ), g∈B(t, u)andξ∈∂_Vϕ(u).

(A5) LetS∈(0, T]and let{u_n}and{ξ_n}be sequences inC([0, S];V)andL^σ(0, S;V^∗), respectively, withσ:= min{2, p}, such thatu_n→ustrongly inC([0, S];V), [u_n(t), ξ_n(t)]∈∂_Vϕfor a.e.

t∈(0, S), and sup_t∈[0,S]ϕ(u_n(t)) +_S

0 |u_n(t)|^p_Hdt+_S

0 |ξ_n(t)|^σ_V∗dt is bounded for alln∈N, and let{g_n}be a sequence inL^p(0, S;V^∗)such thatg_n(t)∈B(t, u_n(t))for a.e.t∈(0, S), g_n →gweakly inL^p(0, S;V^∗).Then,{g_n}is precompact inL^p(0, S;V^∗)andg(t)∈B(t, u(t)) for a.e.t∈(0, S).

(A6) LetS∈(0, T]and letu∈W^1,p(0, S;V)be such thatsup_t∈[0,S]ϕ(u(t))<+∞and suppose that there existsξ∈L^p(0, S;V^∗)such thatξ(t)∈∂_Vϕ(u(t))for a.e.t∈(0, S).Then, there exists a V^∗-valued strongly measurable functiongsuch thatg(t)∈B(t, u(t))for a.e.t∈(0, S). Moreover, the setB(t, u)is convex for allt∈(0, T)andu∈D(B(t,·)).

Now, our result on local (in time) existence is stated as follows:

Theorem 2.1 (Akagi[1]) Letp ∈ (1,∞)andT > 0be given. Suppose that(A1)-(A6)are all satisfied. Then, for all f ∈L^p(0, T;V^∗)andu₀∈D(ϕ), there existsT_∗=T_∗(ϕ(u₀) +|u₀|_H+f_Lp(0,T;V^∗))∈(0, T]such that(CP)admits at least one strong solutionu∈W^1,p(0, T_∗;V)on[0, T_∗].

3 Application to (IBVP)

Applying Theorem 2.1 to (IBVP), we have the following existence result.

Theorem 3.1 (Akagi[1]) LetT >0and suppose that 2≤p < m^∗:=

_mN

N−m if m < N,

+∞ if m≥N and 1< q < m^∗ p + 1.

Then, for allf ∈ L^p(0, T;L^p(Ω))andu₀ ∈ W₀^1,m(Ω), there existsT_∗ =T_∗(ϕ(u₀) +f_Lp(0,T;L^p(Ω)))> 0such that (IBVP)admits at least one solutionu∈W^1,p(0, T_∗;L^p(Ω))on[0, T_∗].

Acknowledgements The author is supported in part by the Shibaura Institute of Technology grant for Project Research (No. 211459 (2006), 211455 (2007)), and the Grant-in-Aid for Young Scientists (B) (No. 19740073), Ministry of Education, Culture, Sports, Science and Technology.

References

[1] G. Akagi, submitted.

[2] G. Akagi and M. ˆOtani, Adv. Math. Sci. Appl.,14, 683–712 (2004).

[3] M. Aso, M. Fr´emond and N. Kenmochi, Nonlinear Anal.60, 1003–1023 (2005).

[4] P. Colli, Japan J. Indust. Appl. Math.9, 181–203 (1992).

[5] M. ˆOtani, J. Diff. Eq.46, 268–299 (1982).

ICIAM07 Contributed Papers 2040048