# The Cauchy Problem for Parabolic Equations with Degeneration

Authors: Ivan Pukal’skii; Bohdan Yashan
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Annotation. For a second-order parabolic equation, the multipoint in time Cauchy problem is considered. The coefficients of the equation and the boundary condition have power singularities of arbitrary order in time and space variables on a certain set of points. Conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight are found.


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## Body

## 1. Introduction

One of the important directions of development of the modern theory of partial differential equations is the study of nonlocal boundary value problems for different types of differential equations and partial differential equation systems and establishing the conditions for their correct solvability. Such problems arise when modeling different phenomena and processes in modern science, quantum mechanics, technology, economics, etc.Multipoint problems for partial differential equations were studied in the papers by Ptashnyk and his disciples. In particular, paper [1] is dedicated to multipoint problems for partial differential equations, not resolved relative to the highest time derivative. The question of the existence and qualitative properties of solutions for equations with limited order of degeneration has been studied in papers [2–4]. The Dirichlet problem with impulse action for a parabolic equation with power singularities of arbitrary order on time and spatial variables is investigated in paper [5]. In paper [6], a multipoint one-sided boundary value problem is studied. In paper [7], presented is the result of research of optimal control of the system described by the problem with an oblique derivative and an integral condition on time variable for parabolic equations with power singularities of arbitrary order. For a second-order parabolic equation, a multipoint (in time) problem with oblique derivative is considered in paper [8]. Conditions for the existence and uniqueness of solution of the posed problem in Hölder spaces with power weight are established.In this paper, we investigate a multipoint time-varying Cauchy problem for a parabolic second-order equation with power singularities and arbitrary order degenerations in the coefficients of spatial and time variables at some set of points. We also find conditions for the existence and uniqueness of the solution of formulated problem in Hölder spaces with power weight.


## 2. Statement of the Problem and Main Result

Letη,t0,t1,⋯,tN,tN+1 be fixed positive numbers, 0≤t0<⋯<tN+1, t0<η<tN+1, η≠tλ, λ∈1,2,⋯,N, and let Ω be some bounded domain in Rn−1, Q0=t,x∣t∈t0;tN+1,x∈Ω∪t,x∣t=η,x∈Rn\Ω¯.Let us consider the problem of finding a functionux,t in the domain Π=t0,tN+1×Rn, which, for t≠tλ, t,x∈Q0 satisfies the equation
(1)∂t−∑i,j=1nAijt,x∂xi∂xj+∑i=1nAit,x∂xi+A0t,xut,x=ft,x,and multipoint conditions for variable t(2)utk+0,x=φkx,k∈0,1,⋯,N.The power singularities of coefficients of the differential equation (1) at the point Pt,x∈Π\Q0 characterize functions s1βi1,t, s2βi2,x:
(3)s1βi1,t=t−ηβi1,t−η≤1,1,t−η≥1,s2βi2,x=ρβi2x,ρx≤1,1,ρx≥1,where ρx=infz∈Ω¯x−z, βiν∈−∞,∞, βν=β1ν,⋯,βnν, ν∈1,2, β=β1,β2.Let us denote thatl, α, q1, q2, γ1, γ2, μj1, μj2 are real numbers, qν≥0, γν≥0, l≥0, μjν≥0, j∈0,1,⋯,n, l is the integer part of l, l=l−l, x1=x11,⋯,xi1,⋯,xn1, x2=x11,⋯,xi−11,xi2,xi+11,⋯,xn1, Pt,x, P1t1,x1, P2t2,x1, Rit2,x2 are arbitrary points of the domain Qk, i∈1,2,⋯,n.LetD be an arbitrary closed domain in Rn, Qk=tk,tk+1×D, Q¯k⊂Πk=tk,tk+1×Rn.We define the functional space in which we study problem (1) and (2).Hlγ;β;q;Π is a set of functions ut,x of space L1Π, which are having continuous partial derivatives in Qk\Q0 of the form ∂ti∂xru, 2i+∣r∣≤l and a finite value of the norm
(4)u;γ;β;q;Πl=supksupQ¯ku≡u,Q0,u;γ;β;q;Πl=supk∑2i+r≤lu;γ;β;q,Qk2i+r+u;γ;β;q;Qkl=supk∑2i+r≤lsupP∈Q¯ks1q1+2i+rγ1,t1×s2q2+2iγ2,x∂ti∂xruP∏j=1ns1−rjβj1,ts2rjγ2−βj2,x+supk∑2i+r=l∑ν=1nsupP2,Rν⊂Q¯ks1q1+lγ1,t2s2q2+2iγ2,x~×∏j=1ns1−rjβj1,t2s2rjγ2−βj2,x~∂ti∂xruP2−∂ti∂xruRν×xν1−xν2−ls1lβν1,t2s2lγ2−βν2,x~+supP1,P2⊂Q¯ks1q1+lγ1,t~s2q2+2i+lγ2,x1×∏j=1ns1−rjβj1,t~s2rjγ2−βj2,x1t1−t2−1/2∂ti∂xruP1−∂ti∂xruP2.Marked here,r=r1+r2+⋯+rn, s1q,t~=mins1q,t1,s1q,t2, s2q,x~=mins2q,x1,s2q,x2.We assume that the initial problems (1) and (2) satisfy the following conditions:
(a)
For the arbitrary vectorξ=ξ1,⋯,ξn, ∀t,x∈Π\Q0, the following inequality(5)c1ξ2≤∑i=1ns1βi1,ts1βj1,ts2βi2,xs2βj2,xAijt,xξiξj≤c2ξ2is true, where c1,c2 are fixed positive constants and s1μi1,ts2μi2,xAi∈Hαγ;β;0;Π, s1μ01,ts2μ02,xA0∈Hαγ;β;0;Π, A0≥−a, a>0, s1βi1,ts1βj1,ts2βi2,x××s2βj2,xAij∈Hαγ;β;0;Π, γν=maxmaxi1+βiν,maxiμiν−βiν,μ0ν/2, ν∈1,2.
(b)
Functionsf∈Hαγ;β;μ0;Π, φ0∈H2+αγ~;β~;0;Rn, φλ∈H2+αγ~;β~;μ0;Π∩t=tλ, γ~=0;γ2, β~=0;β2Let us formulate the main result of the paper.Theorem 1.
Let conditions (a) and (b) be satisfied for problem (1) and (2). Then, there exists a unique solution of problem (1) and (2) in the space H2+αγ;β;0;Π, and the following estimate is correct:
(6)u;γ;β;0;Π2+α≤csupk∈0,1,⋯,Nφk;γ,β,0;Π∩t=tk2+α+f;γ,β,μ0;Πkα.
To study problem (1) and (2), we construct a sequence of solutions of problems with a smooth coefficient limit value of which is the solution of problem (1) and (2).


## 3. Evaluation of Solutions of Problems with Smooth Coefficients

LetΠmk=Πk∩t,x∈Πk∣s11,t≥m1−1,s21,x≥m2−1, m=m1,m2, m1>1, m2>1 be a sequence of domains that for m1→∞, m2→∞ converges to Πk.In the domainΠ, we consider the problem of finding the functions umt,x that satisfy the equations
(7)L1umt,x=∂t−∑ij=1naijt,x∂xi∂xj+∑i=1nait,x∂xi+a0t,xumt,x=fmt,x,and the time variable t condition
(8)umtk+0,x=φmkx,x∈Qk∩t=tk.Here, the coefficientsaij, ai, a0, and functions fm, φmk, into the domains Πmk coincide with Aij, Ai, A0, f, φk, respectively, and in the domains Π\Πmk are continuous prolongations of coefficients Aij, Ai, A0 and functions f, φk from domains Πmk into domains ΠΠmk with preservation of their smoothness and norm ([9], p. 82).To solve problem (7) and (8), we have a correct theorem.Theorem 2.
Letumt,x be the classical solutions of problem (7) and (8) in the domain Π and let conditions (a) and (b) be satisfied. Then, for umt,x, the following estimate
(9)um;Πk0≤φmk;Πk∩t=tk0+fma0−1Πk0is true.Proof.
LetmaxQ¯kumt,x=umM1. If M1∈Qk, then at the point M1, the following correlations
(10)∂tumM1≥0,∂xiumM1=0,∑i,j=1naijM1∂xi∂xjumM1≤0are true and equation (7) is satisfied. Considering (10) and equation (7) at the point M1, the inequality
(11)umM1≤supQ¯kfa0−1is correct.
LetminQ¯kumt,x=umM2. If M2∈Qk, then at the point M2, the following correlations
(12)∂tumM2≤0,∂xiumM2=0,∑i,j=1naijM2∂xi∂xjumM2≥0are true and equation (7) is satisfied. Considering correlation (12) and equation (7) at the point M2, we have
(13)umM2≥infQ¯kfa0−1.
In the case ofM1∈Qk∩t=tk or M2∈Qk∩t=tk from condition (8), we obtain
(14)um≤φmk;Qk∩t=tk0.
Considering inequality (11), (13), and (14), we obtain
(15)um;Πk≤fma0−1;Πk0+φmk;Πk∩t=tk0.
The theorem is proved.Now, we find estimates of the derivatives of solutionsumt,x. In the space ClΠ, we introduce a norm um;γ;β;q;Πl which is equivalent, at fixed m1, m2, to the Hölder norm, which is defined by the same way as the norm u;γ;β;q;Πl; only, instead of functions s1q1,t and s2q2,x, we take d1q1,t and d2q2,x, respectively,
(16)d1q1,t=maxs1q1,t,m1−q1,q1≥0,mins1q1,t,m1−q1,q1<0,d2q2,x=maxs2q2,x,m2−q2,q2≥0,mins2q2,x,m2−q2,q2<0.Theorem 3.
Let conditions (a) and (b) be satisfied. Then, for the solution of problem (7) and (8), the estimate
(17)um;γ;β;0;Πk2+α≤cf;γ;β;μ0;Πkα+φk;γ;β;0;Πk∩t=tk−12+αis true.Proof.
Using the definition of the norm and interpolation inequalities from [9], we have
(18)um;γ;β;0;Πk2+α≤1+εαum;γ;β;0;Πk2+α+Cεum;Πk0,where ε is an arbitrary real number ε∈0,1. Hence, it is sufficient to estimate the seminorm um;γ;β;0;Πk2+α. As follows from the definition of seminorm, there exist in Πk points P1,P2,Rν, for which one of the inequalities
(19)12um;γ;β;0;Πk2+α≤Eμ,μ∈1,2,is true, where
(20)E1≡∑2i+r=2∑ν=1nd12γ1,t2d22iγ2,x~∏i=1nd1−riβi1,t2×d2riγ2−βi2,x~∂ti∂xrumP2−∂ti∂xrumRνxν1−xν2−α×d1αβν1,t2d2αγ2−βν2,x~,E2≡∑2i+∣r∣=2d12+αγ1,t~d2riγ2−βi2,x1∏i=1nd1−riβi1,t~×d2riγ2−βi2,x1t1−t2−α/2∂ti∂xrumP1−∂ti∂xrumP2,2s+r=2.
Ifxν1−xν2≥εn−1/4d1γ1,t~d2γ2−βν2,x~≡T1, and ε1 is an arbitrary real number from (0,1), then
(21)E1≤2ε1−αum;γ;β;0;Πk2.
Ift1−t2≥ε12/16d12γ1,t~d22γ2,x~≡T2, then
(22)E2≤2ε1−αum;γ;β;0;Πk2.
Applying the interpolation inequalities to (21) and (22), we find
(23)Eμ≤εαum;γ;β;0;Πk2+α+cεum;Πk0.
Letxν1−xν2≤T1, and let t1−t2≤T2. We assume that xν1−ξν≥4T1, ξ∈∂D and
(24)d2γ2,x~=mind2γ2,x1,d2γ2,x2≡d2γ2,x1,d1γ2,t~=mind1γ1,t1,d1γ1,t2=d1γ1,t1,
P1t1,x1∈Πk, k∈0,1,⋯,N. In the domain Qk, we write the problem (7) and (8) in the form
(25)∂t−∑ij=1naijP1∂xi∂xjum=∑ij=1naijP−aijP1∂xi∂xjum−∑i=1naiP∂xium−a0Pum+fmt,x=Fmt,x,(26)umtk+0,x=Φmktk,x.
LetΠτk be a domain from Qk, Πτk=t,x∈Qk,∣xν−xν1≤τT1,ν∈1,2,t−t1≤τ2T2. Performing the substitution,
(27)umt,x=vmt,y,xν=d1βν1,t1d2βν2,x1yν,in problem (25) and (26), we obtain
(28)L2vmt,y≡∂t−∑ij=1naijP1d1βi1,t1d1βj1,t1×d2βi2,x1d2βj2,x1∂yi∂yjvm=Fmt,y~,vmtk+0,y~=Φmktk,y~,where y~=d1−β11,t1d2−β12,x1x1,⋯,d1−βn1,t1d2−βn2,x1xn.We denote(29)yν1=d1−βν1,t1d2−βν2,x1xν1,Vτk=t,y,t−t1≤τ2T2,yν−yν1≤τnT2,and choose a thrice differentiable function ψt,y, which satisfies the conditions
(30)ψt,y=1,t,y∈V1/2k,0≤ψt,y≤1,0,t,y∈V3/4k,∂ti∂xrψ≤crid1−2i+rγ1;t1××d2−2i+rγ2,x1.We denote the functionZmt,y=vmt,yψt,y which is a solution of the Cauchy problem
(31)L3Zmt,y=∑ij=1naijP1d1βi1,t1d1βj1,t1d2βi2,x1d2βj2,x1×∂yivm∂yjψ+∂yjvm∂yiψ+vm∂yi∂yjψ−vm∂tψ+ψFmt,y~=Fm1t,y~,ψ,(32)Zmtk+0,x=Φmktk,y~ψtk,y.On the basis of Theorem 5.3 from ([2], p. 364), for the solution of problem (31) and (32) and arbitrary points M1,M2⊂V1/2k inequalities
(33)d−αM1,M2∂ti∂xrvmM1−∂ti∂xrvmM2≤cFm1CαV3/4+ψΦmkC2+αV3/4∩t=tkare true, where 2i+r=2, dM1,M2 is the parabolic distance between the points M1 and M2. With regard to the properties of function ψt,y, we find estimates of the norms of expressions Fm1 and ψΦmk:
(34)Fm1CαV3/4≤cd1−2+αγ1,t1d2−2+αγ2,x1×vm;V3/4k0+vm;γ;0;0;V3/4k2+Fm;γ;0;2γ;V3/4kα,(35)ψΦmkC2+αV3/4∩t=tk≤cd1−2+αγ1,t1d2−2+αγ2,x1×Φmk;γ;0;V3/4k2+α.The definition of the spaceHlγ;β;q;Q implies the satisfaction of inequalities
(36)c1vm;γ;0;0;V3/4kl≤um;γ;β;0;Π3/4kl≤c2vm;γ;0;0;V3/4kl.Substituting (34) and (35) into (33) and returning to the variables t,x, we obtain inequalities
(37)Eμ≤c1Fm;γ;β;2γ;Π3/4kα+Φmk;γ;β;0;Π3/4k∩t=tk2+α+um;Π3/4k0+um;γ;β;0;Π3/4k2.Given the interpolation inequalities and estimates of the norm of each additive of the expressionsFm, Φmk, we obtain the inequalities
(38)Eμ≤εαn+2+ε1Cn2um;γ;β;0;Π3/4k2+α+c2um;Π3/4k0+c3Fm;γ;β;2γ;Π3/4kα+Φmkγ;β;0;Π3/4k∩t=tk2+α.Using inequalities (23) and (37) and choosing ε1 and ε sufficiently small, we obtain the estimate
(39)um;γ;β;0;Πk2+α≤cFm;γ;β;2γ;Πkα+Φmkγ;β;0;Πk∩t=tk2+α.Given the values of the expressionFmt,x and Φmktk,x, for k=1,2,⋯,N, we have
(40)Fm;γ;β;2γ;Πkα≤cfm;γ;β;μ0;Πkα,Φmk;γ;β;0;Π∩t=tk2+α≤cφmk;γ;β;0;Π∩t=tk2+α.Because(41)fm;γ;β;μ0;Πkα≤cf;γ;β;μ0;Πkα,φmk;γ;β;0;Πk∩t=tk2+α≤cφk;γ;β;0;Πk∩t=tk2+α,then, given the estimate (9) and inequalities (39) and (40) for k=0,1,⋯,N, the estimate (17) is true. The theorem is proved.Proof of Theorem 4.
The right-hand side of inequality (17) is independent of m, and the sequences
(42)um0≡um,um1≡d1γ1−βi1,td2γ2−βi2,x∂xium,um2≡d12γ1,td22γ2,x∂tum,um3=d1γ1−βi1,td2γ2−βi2,xd1γ1−βj1,td2γ2−βj2,x∂xi∂xjumare uniformly bounded and equicontinuous in Qk. According to Arcel’s theorem, there exist subsequences umjμ which are uniformly convergent to u0μ in Qk for mj→∞, μ∈0,1,2,3. Passing to the limit as mj→∞ in problem (7) and (8), we obtain that ut,x=u00 is the unique solution of problem (1) and (2) and u∈H2+αγ;β;0;Π. The theorem is proved.


## 4. Conclusions

The necessary and sufficient conditions for the existence of the unique solution of a multipoint problem for parabolic equations with degeneration are established. Estimates of derivatives of the solution of the problem in the Hölder spaces with power weight are found. The order of the degree weight depends on the power of the degree features of the coefficients of the equation.



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