SHORT-SS4: Verified computations for solutions to semilinear parabolic equations using the evolution operator

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1 SHORT-SS4: Verified computations for solutions to semilinear parabolic equations using the evolution operator Akitoshi Takayasu 1, Makoto Mizuguchi, Takayuki Kubo 3, and Shin ichi Oishi 4 1 Research Institute for Science and Engineering, Waseda University Graduate School of Fundamental Science and Engineering, Waseda University 3 Institute of Mathematics, University of Tsukuba 4 Department of Applied Mathematics, Waseda University and CREST, JST takitoshi@aoniwasedajp Abstract This article presents a theorem for guaranteeing existence of a solution for an initial-boundary value problem of semilinear parabolic equations The sufficient condition of our main theorem is derived by a fixed-point formulation using the evolution operator We note that the sufficient condition can be checked by verified numerical computations 1 Introduction Let J :=, t 1 ] 0 < t 1 < be a time interval and Ω a convex polygonal domain in IR In this article we consider the following initial-boundary value problems of semilinear parabolic equations: t u u = fu in J Ω, ut, x = 0 on J Ω, 1 u, x = u 0 x in Ω Here, t u = u t, = denotes the Laplacian, the domain of the x 1 x Laplacian is D = H Ω H0 1 Ω, fu is a real-valued function in J Ω such that f : H0 1 Ω L Ω is a twice Fréchet differentiable nonlinear mapping for t J, and u 0 H0 1 Ω is an initial function Let τ := t 1 The main aim of this article is to present Theorem 1 for proving existence and local uniqueness of a solution to 1 in a neighborhood of an approximate solution This approximate solution consists of two numerical solutions Let V h be a finite dimensional subspace of DA For two numerical solutions û 0, û 1 V h, we define the approximate solution ωt as ωt = û 0 ϕ 0 t û 1 ϕ 1 t, t J, where ϕ i t i = 0, 1 is a linear Lagrange basis such that ϕ i t j = δ ij δ ij is a Kronecker s delta for j = 0, 1 The evolution operator is introduced by Tanabe and Sobolevskii [1, ] Using the evolution operator, studies of parabolic equation have been developed in the field of mathematical analysis cf [3, 4]

2 In this article, we present a fixed-point form by using the evolution operator Existence of its fixed-point is equivalent to that of the mild solution to 1 We then derive a sufficient condition for verifying existence of the fixed-point By numerically checking whether the sufficient condition holds, existence and local uniqueness of the mild solution to 1 are proved Fixed-point formulation Let us start from the following fact: the mild solution u of 1 exists if and only if the function z = u ω is the mild solution of t z z = fz ω t ω ω in J Ω, zt, x = 0 on J Ω, z, x = u 0 x û 0 x in Ω Suppose that z = e σt t0 v holds for a certain σ > 0 Then v is a solution of the following equation: t v Atv = gv in J Ω, vt, x = 0 on J Ω, 3 v, x = u 0 x û 0 x in Ω, where At = σ f [ωt], gv = e {f σt t0 ω e σt t0 v } fω f [ωt]e σt t0 v fω t ω ω holds The operator f [ωt] : H 1 0 Ω L Ω denotes a Fréchet derivative of f at ωt for t J We furthermore assume that f [ωt] is a symmetric operator for t J From the definition of ω in, the domain of At becomes D At = D for each t J DAt is independent of t J Let us fix µ > 0 We define a norm of V = H 1 0 Ω as ϕ V = ϕ L µ ϕ L 1/ for ϕ V We determine the σ > 0 such that σ f [ωt] µ ae Ω for t J It then follows and Atu, v L = u, v L σ f [ωt]u, v L = u, v L µu, v L σ f [ωt] µu, v L 1 C σ C µ u V v V Atu, u L = u, u L σ f [ωt]u, u L u V,

3 where C σ = sup x Ω σ f [ωt] µ, C µ > 0 such that ϕ L C µ ϕ V This yields DAt 1/ = V, ie, the following holds for ϕ V : ϕ V At 1/ ϕ L M ϕ V, M = 1 C σ C µ 1/ For each t J, At thus becomes the sectorial operator and generates a holomorphic semigroup { e sat} s 0 over L Ω The eigenvalue of At for t J is bounded below by λ A = λ min µ > 0, where λ min denotes the minimum eigenvalue of Therefore, the operator At becomes a symmetric positive operator Additionally, for t, s J there exists C > 0 and α > 0 such that AtAs 1 I L,L = At AsAs 1 L,L C t s α, where L,L denotes the operator norm over L Ω From the above facts it is well-known [1 4] that the operator At generates an evolution operator {Ut, s} t0 s t t 1 on L Ω The evolution operator is described by Ut, s = e t sas s e t rar Rr, sdr s r t t 1, where Rt, s is the solution of the following integral equation: Rt, s = R 1 t, s R 1 t, rrr, sdr, R 1 t, s = At Ase t sas By using the evolution operator {Ut, s} t0 s t t 1, we define a nonlinear operator T : CJ; V CJ; V as T v := Ut, v s 4 Ut, sgvsds s t t 1 5 If v satisfies the fixed-point form v = T v in CJ; V, then there exists a solution of 3 that is described by the evolution operator In the following we derive a sufficient condition for verifying existence of the solution to 3 If this sufficient condition holds, existence of the mild solution to 1 is also proved 3 Main theorem Let us define a function space { } X σ := v C J; V : sup e σt t0 vt V < t J with the norm v Xσ := sup t J e σt t0 vt V The following theorem gives a sufficient condition for guaranteeing existence and local uniqueness of a mild solution to 1 in B J ω, ρ := { u C J; V : u ω CJ;V ρ }

4 Theorem 1 Assume that û 0 V h satisfies u 0 û 0 H 1 0 ε 0 and 0 σ < λ A holds Assume that ω satisfies the following estimate: t ω ω fω CJ;L Ω δ Assume also that there exists a monotonically non-decreasing function L ω : [0, [0, corresponding to the first Fréchet derivative of f : H 1 0 Ω L Ω such that f [ω h] f [ω]ϕ CJ;L Ω L ω ρ ϕ CJ;V, ϕ C J; V, where h X σ satisfying h Xσ ρ for a certain ρ > 0 If M O 1 τ ε 0 π eλ A σ erf λa στ 1 O τ L ω ρρ δ < ρ 6 holds, then the mild solution ut := ut,, t J, of 1 uniquely exists in the ball B J ω, ρ Here, O 1 τ and O τ in 6 are given by Cω O 1 τ = C µ e τ 1 sinh Cω τ and O τ = C ω τ sinh Cω τ, respectively, if R 1 t, s in 4 satisfies R 1 t, s L,L C ωt se t sλ A Before we sketch a proof of the main theorem, some lemmas are necessary Lemma 1 If R 1 t, s in 4 satisfies R 1 t, s L,L C ωt se t sλ A, it follows Rt, s L,L C ω sinh Cω t s e t sλ A Lemma For the evolution operator {Ut, s} t0 s t t 1 generated by At and v = u 0 û 0, the following estimate holds: Ut, v V Me t t0λ A O 1 τ e 1 t t0λ A ε 0 Lemma 3 For the evolution operator {Ut, s} t0 s t t 1 and gv in 3, the following estimate holds: generated by At Ut, sgvs V e 1 t s 1 e 1 t sλ A gvs L 1 O τ Proofs of these lemmas are omitted for lack of space Sketch of the proof For ρ > 0 let Z = {v X σ : v Xσ ρ} Let us consider the fixed-point form 5 On the basis of Banach s fixed-point theorem, we give a sufficient condition of T having a fixed-point in Z First, we derive a condition guaranteeing T Z Z For v Z, Lemma 1 and gives Me t λ A O 1 τ e 1 t λ A T vt V e 1 t s 1 e 1 t sλ A gvs L 1 O τ ds ε 0

5 It follows e σt t0 T vt V Me t t0λa σ O 1 τ e 1 t t0λ A σ ε 0 e 1 t s 1 e 1 t sλ A σ e σs gvs L 1 O τ ds 7 From 3 and the assumptions of the theorem, we have e σs t0 gvs L f ωs e σs t0 vs fωs f [ωs]e σs t0 vs fωs s ωs ωs L L ω ρρ δ The upper bound of 7 with respect to t J is given by T v Xσ M O 1 τ ε 0 π eλ A σ erf λa στ 1 O τ L ω ρρ δ From 6 T v Xσ < ρ holds Namely, we obtain T v Z Next, under the assumptions of the theorem, we show that T is a contraction mapping on Z For v 1, v Z, we have T v 1 T v Xσ π eλ A σ erf λa στ 1 O τ L ω ρρ v 1 v Xσ The assumption 6 also implies π eλ A σ erf λa στ 1 O τ L ω ρρ < 1 Therefore, T becomes a contraction mapping on Z Banach s fixed-point theorem asserts that there uniquely exists a fixed-point v = T v in Z It yields that the mild solution of 1 uniquely exists in the ball B J ω, ρ References 1 H Tanabe, On the equations of evolution in a Banach space, Osaka Mathematical Journal, 1: 1960, pp P E Sobolevskii, On equations of parabolic type in Banach space with unbounded variable operator having a constant domain, Akad Nauk Azerbaidzan SSR Doki, 17: in Russian 3 A Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, H Fujita, N Saito, T Suzuki, Operator theory and numerical methods, North Holland, 001 L