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European Journal of Applied Sciences – Vol. 12, No. 6

Publication Date: December 25, 2024

DOI:10.14738/aivp.126.17983.

Ayala, Y. S. S (2024). Wellposedness of a Cauchy Problem Associated to the Even Order Equation. European Journal of Applied

Sciences, Vol - 12(6). 512-530.

Services for Science and Education – United Kingdom

Wellposedness of a Cauchy Problem Associated to the Even Order

Equation

Yolanda Silvia Santiago Ayala

ORCID: 0000-0003-2516-0871

Universidad Nacional Mayor de San Marcos,

Fac. de Ciencias Matemáticas, Av. Venezuela Cda. 34 Lima-PERU

ABSTRACT

In this article we prove that the Cauchy problem associated to n-th order equation

in periodic Sobolev spaces is globally well posed when n is an even number

multiple of four. We do this in an intuitive way using Fourier theory and in a fine

version using semigroups theory. Finally, we demonstrate the dissipative property

of the Cauchy problem using differential calculus in Hper

s

.

Keywords: Semigroups theory, even order equation, existence of solution, dissipative

property, Periodic Sobolev spaces, Fourier Theory.

MSC 2010: 35G10, 35Q53, 47D06, 35B40

INTRODUCTION

We study the problem:

(P1): ∂tu + ∂x

nu = 0 in Hper

s−n

, with u(0) = φ ∈ Hper

s

,

considering s a real number, n is an even number multiple of four and denoting by Hper

s

to the

periodic Sobolev space. This problem was proposed in [9], specifically in remark 4.3. In [9],

the case n=3 was studied and some comments on the possibility of its generalization were

given.

Also we can cite [6], where we find some works related to the model (P1) coupled to

Kuramoto-Sivashinski equation, and [1] where our model is stated among the proposed

problems. In these additional references we have motivation to study the problem and

inspiration with the ideas we find there.

We also cite some works about existence by semigroups [2], [3], [4] and take support in some

results of [5].

Our article is organized as follows. In section 2, we indicate the methodology used and cite the

references used. In section 3, we prove that problem (P1) is well posed. Moreover, we

introduce a family of operators that form a semigroup of class C0 to state the result Theorem

3.3 and prove it in a fine version. In section 4, we study the dissipative property of the

homogeneous problem (P1) and applications. Finally, in section 5, we give the conclusions of

our study.

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Ayala, Y. S. S (2024). Wellposedness of a Cauchy Problem Associated to the Even Order Equation. European Journal of Applied Sciences, Vol - 12(6).

512-530.

URL: http://dx.doi.org/10.14738/aivp.126.17983

METHODOLOGY

As a theorical framework in this article we use [6]. Also, we use the references [1], [8], [6] and

[7] for the Fourier theory in periodic Sobolev spaces, and differential and integral calculus in

Banach spaces.

THE PROBLEM (P1) IS WELL POSED

We prove that (P1) is well posed. Also, we introduce a family of operators that form a

semigroup of class Co, as we make it in Theorem 3.2. Finally, we state the Theorem 3.3 whose

content is a fine version of Theorem 3.1 based on the semigroup {S(t)}t≥0 .

Theorem 3.1

Let s a fixed real number, n an even number multiple of four and the problem

(P1) |

u ∈ C([0, ∞), Hper

s

)

∂tu + ∂x

nu = 0 ∈ Hper

s−n

u(0) = φ ∈ Hper

s

then (P1) is globally well posed, that is ∃! u ∈ C([0, ∞), Hper

s

) ∩ C

1

((0, ∞), Hper

s−n

) satisfying

equation (P1) so that the application : φ → u , which to every initial data φ assigns the solution u

of the IVP (P1), is continuous. That is, for φ and φ̃ initial data close in Hper

s

, their corresponding

solutions u and ũ respectively, are also close in the solution space.

Also,

ǁu(t) − ũ(t)ǁs  ǁφ − φ̃ ǁs , ∀t ∈ [0, ∞)

and

supt∈[0,∞) ǁu(t) − ũ(t)ǁs = ǁφ − φ̃ ǁs .

Moreover, the solution u satisfies

u(t) ∈ Hper

r

, ∀t ∈ [0, ∞) , ∀r ≤ s

with

ǁu(t)ǁs  ǁφǁs and ǁu(t)ǁr ≤ ǁφǁs , ∀r < s and t ∈ [0, ∞).

The application: φ → ∂tu, wich for every initial data φ assigns the derivative of solution u of the

IVP (P1) is continuous. That is, for φ and φ̃ initial data close in Hper

s

their corresponding ∂tu and

∂tũ , respectively, are also close in the solution space. Also, the following inequalities are verified

ǁ∂tu(t) − ∂tũ (t)ǁs−n ≤ ǁφ − φ̃ǁs, ∀t ∈ (0, ∞) ,

supt∈(0,∞), ǁ∂tu(t) − ∂tũ (t)ǁs−n ≤ ǁφ − φ̃ǁ.

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024

Moreover, ǁ∂tu(t)ǁs−n ≤ ǁφǁs, ∀t ∈ (0, ∞).

Proof:

We prove it in the following way.

1. First, we obtain the candidate to the solution. In order to get it we apply the Fourier

transformation to the equation

∂tu = −∂x

nu

and using (ik)

n = k

n

, ∀ k ∈ Z, we have

∂tû = −(ik)

nû

= −k

nû

which for every k is an ODE with initial data û(k, 0) = φ̂(k).

Thus, solving the IVP’s

(Ωk

) |

û ∈ C([0, ∞),ls

2

(Z))

∂tû(k,t) = −k

n û(k,t)

û(k, 0) = φ̂(k)

we obtain

û(k,t) = e

−k

ntφ̂(k),

from which we get our candidate to the solution:

u(t) = ∑ û(k,t)φk

+∞

k=−∞

= ∑ e

−k

ntφ̂(k)φk (3.1)

+∞

k=−∞

here we are denoting φk

(x) = e

ikx for xR.

2. Second, we prove:

u(t) ∈Hper

s

and ǁu(t)ǁs  ǁφǁs (3.2)

In effect, let t ∈ (0,), φ ∈ Hper

s

, we have