Degenerate abstract parabolic equations with variable coefficients are
studied. Here the boundary conditions are nonlocal. The maximal regularity
properties of solutions for elliptic and parabolic problems and Strichartz type
estimates in mixed $L_{p}$ spaces are obtained. Moreover, the existence and
uniqueness of optimal regular solution of mixed problem for nonlinear parabolic
equation is established. Note that, these problems arise in fluid mechanics and
environmental engineering

Sahmurova, Aida

Shakhmurov, Veli

English

arXiv

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary
conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems
and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness
of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these
problems arise in fluid mechanics and environmental engineering

Shakhmurov, Veli, B.

Khazar University Institutional Repository

CARPATHIAN J. MATH.34 (2018), No. 2, 247 - 254Online version at http://carpathian.ubm.roPrint Edition: ISSN 1584 - 2851 Online Edition: ISSN 1843 - 4401Mixed problems for degenerate abstract parabolicequations and applicationsVELI. B. SHAKHMUROV1,2 and AIDA SAHMUROVA3ABSTRACT. Degenerate abstract parabolic equations with variable coefficients are studied. Here the boun-dary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problemsand Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniquenessof optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, theseproblems arise in fluid mechanics and environmental engineering.1. INTRODUCTIONIn this work, the boundary value problems (BVPs) for parameter dependent degene-rate differential-operator equations (DOEs) are considered. Namely, linear equations andboundary conditions contain small parameters and are degenerated in some part of boun-dary. These problems have numerous applications in PDE, pseudo DE, mechanics andenvironmental engineering. The BVP for DOEs have been studied extensively by manyresearchers (see e.g. [1–11] and the references therein). The maximal regularity proper-ties for DOEs in Banach space valued function class are investigated e.g. in [2, 3, 5–10].Nonlinear DOEs studied e.g. in [6, 9].The main objective of the present paper is to discuses the initial and nonlocal BVP forthe following nonlinear degenerate parabolic equation(1.1)∂u∂t+n∑k=1ak (x)∂[2]u∂x2k+B((t, x, u,D[1]u))u = F(t, x, u,D[1]u),where ak are complex valued functions, B and F are nonlinear operators in a Banachspace E andD[1]u =(∂[1]u∂x1,∂[1]u∂x2, ...,∂[1]u∂xn), x = (x1, x2, ..., xn) ∈ G =n∏k=1(0, bk) ,D[i]ku = u(i)k =∂[i]u∂xik=(xαkk∂∂xk)iu (x) , 0 ≤ αk < 1.First all of, we consider the nonlocal BVP for the degenerate elliptic DOE with smallparameters(1.2)n∑k=1ak (x)∂[2]u∂x2k+A (x)u+ λu+n∑k=1Ak (x)∂[1]u∂xk= f (x) ,Received: 05.09.2016. In revised form: 06.07.2018. Accepted: 13.07.20182010 Mathematics Subject Classification. 35A01 35J56 35K51 47G40.Key words and phrases. differential-operator equations, degenerate PDE, semigroups of operators, nonlinear pro-blems, separable differential operators, positive operators in Banach spaces.Corresponding author: Veli. B. Shakhmurov; veli.sahmurov@okan.edu.tr247248 Veli. B. Shakhmurov and Aida Sahmurovawhere ak are complex-valued functions, λ is a complex parameters, A (x) and Ak (x) arelinear operators.We prove that for f ∈ Lp (G;E), |arg λ| ≤ ϕ, 0 < ϕ ≤ pi and sufficiently large |λ| ,problem (1.2) has a unique solution u ∈ W [2]p,α (G;E (A) , E) and the following coerciveuniform estimate holdsn∑k=12∑i=0|λ|1− i2∥∥∥∥∂[i]u∂xik∥∥∥∥Lp(G;E)+ ‖Au‖Lp(G;E) ≤ C ‖f‖Lp(G;E) .Then the above result is used to prove the well-posedeness of initial BVP (IBVP) andthe uniform Strichartz type estimate for the solution the degenerate abstract parabolicequation with parameters(1.3)∂u∂t+n∑k=1ak (x)∂[2]u∂x2k+A (x)u = f (x, t) , t ∈ (0, T ) , x ∈ G.Finally, via maximal regularity properties of (1.3) and contraction mapping argument, theexistence and uniqueness of solution of the problem (1.3) is derived.Note that, the equation and boundary conditions are degenerated with the differentrate at different boundary edges, in general.Let γ = γ (x) be a positive measurable function on Ω ⊂ Rn and E be a Banach space.Let Lp,γ (Ω;E) denote the space of strongly measurable E-valued functions defined on Ωwith the norm‖f‖Lp,γ = ‖f‖Lp,γ(Ω;E) =(∫‖f (x)‖pE γ (x) dx) 1p, 1 ≤ p <∞.For γ (x) ≡ 1 we will denote these spaces by Lp (Ω;E) .LetE0 andE be two Banach spaces andE0 is continuously and densely embeds intoE.Let us consider the Sobolev-Lions type space Wmp,γ (a, b;E0, E) , consisting of all functionsu ∈ Lp,γ (a, b;E0) that have generalized derivatives u(m) ∈ Lp,γ (a, b;E) with the norm‖u‖Wmp,γ = ‖u‖Wmp,γ(a,b;E0,E) = ‖u‖Lp,γ(a,b;E0) +∥∥∥u(m)∥∥∥Lp,γ(a,b;E)<∞.Let γ = γ (x) be a positive measurable function on (0, 1) andW [m]p,γ = W[m]p,γ (0, 1;E0, E) = {u : u ∈ Lp (0, 1;E0) ,u[m] ∈ Lp (0, 1;E) , ‖u‖W [m]p,γ = ‖u‖Lp(0,1;E0) +∥∥∥u[m]∥∥∥Lp(0,1;E)<∞}.Letαk (x) = xαkk , α = (α1, α2, ..., αn) .Consider E-valued weighted space defined byW [m]p,α (G,E (A) , E) = {u;u ∈ Lp (G;E0) ,∂[m]u∂xmk∈ Lp (G;E) ,‖u‖W[m]p,α= ‖u‖Lp(G;E0) +n∑k=1∥∥∥∥∂[m]u∂xmk∥∥∥∥Lp(G;E)<∞}.Mixed problems for degenerate abstract parabolic equations and applications 2492. DEGENERATE ABSTRACT ELLIPTIC EQUATIONSConsider the BVP for the following degenerate partial DOE with parameters(2.4)n∑k=1ak (xk)∂[2]u∂x2k+A (x)u+ λu+n∑k=1Ak (x)∂[1]u∂xk= f (x) ,Lkju =mkj∑i=0αkjiu[i]xk(Gk0) + βkjiu[i]k (Gkb) = 0, j = 1, 2,where ak are complex-valued functions, A (x) and Ak (x) are linear operators, u = u (x),αkji, βkji are complex numbers, λ is a complex parameter, mkj ∈ {0, 1},x = (x1, x2, ..., xn) ∈ G =n∏k=1(0, bk) ,Gk0 = (x1, x2, ..., xk−1, 0, xk+1, ..., xn) , pk ∈ (1,∞) ,Gkb = (x1, x2, ..., xk−1, bk, xk+1, ..., xn) ,x(k) = (x1, x2, ..., xk−1, xk+1, ..., xn) ∈ Gk =∏j 6=k(0, bj) .Consider the principal part of (2.4), i.e., consider the problem(2.5)n∑k=1ak (xk)∂[2]u∂x2k+A (x)u+ λu = f (x) ,mkj∑i=0αkjiu[i]xk(Gk0) + βkjiu[i]k (Gkb) = 0, j = 1, 2.Theorem 2.1. Assume;(1) E is an UMD spacethe Banach space, 0 ≤ αk < 1 − 1pk , pk ∈ (1,∞), αkmk1 6= 0,βkmk2 6= 0;(2) A (x) is a uniformly R-positive operator in E, A (x)A−1 (x¯) ∈ C (G¯;L (E)), x ∈ G;(3) ak ∈ C(m) ([0, bk]) and ak (xk) < 0 for xk ∈ [0, bk] ;(4) ak (Gj0) = ak (Gjb) , A (Gj0)A−1 (x0) = A (Gjb)A−1 (x0), k, j = 1, 2, ..., n;(5) ηk = (−1)m1 αk1βk2 − (−1)m2 αk2βk1 6= 0.First, we prove the separability properties of the problem (2.5):Then, problem (2.5) has a unique solution u ∈W [2]p,α (G;E (A) , E) for f ∈ Lp (G;E), |arg λ| ≤ϕ with sufficiently large |λ| and the following coercive uniform estimate holds(2.6)n∑k=12∑i=0|λ|1− i2∥∥∥∥∂[i]u∂xik∥∥∥∥Lp(G;E)+ ‖Au‖Lp(G;E) ≤ C ‖f‖Lp(G;E) .Proof. Consider the BVP(2.7) (L+ λ)u = a1 (x1)D[2]x1u (x1) + (A (x1) + λ)u (x1) = f (x1) ,L1ju = 0, j = 1, 2, x1 ∈ (0, b1) ,where L1j are boundary conditions of type (2.5) considered on (0, b1) . By virtue of [7],problem (2.7) has a unique solution u ∈ W [2]p,α1 (0, b1;E (A) , E) for f ∈ Lp1 (0, b1;E),|arg λ| ≤ ϕ with sufficiently large |λ| and the coercive uniform estimate holds250 Veli. B. Shakhmurov and Aida Sahmurova2∑j=0|λ|1− j2∥∥∥u[j]∥∥∥Lp1 (0,b1;E)+ ‖Au‖Lp1 (0,b1;E) ≤ C ‖f‖Lp1 (0,b1;E) .Now, let us consider the following BVP(2.8)2∑k=1ak (xk)D[2]ku (x1, x2) +A (x1, x2)u (x1, x2) + λu (x1, x2) = f (x1, x2) ,Lk1u = 0, Lk2u = 0, k = 1, 2, x1, x2 ∈ G2 = (0, b1)× (0, b2) .Let α (2) = (α1, α2). Since Lp (0, b2;Lp (0, b1) ; E) = Lp (G2;E) , the BVP (2.8) can beexpressed asa2D[2]2 u (x2) + (B (x2) + λ)u (x2) = f (x2) , L2ju = 0, j = 1, 2,for x1 ∈ (0, b1), where B is a differential operator in Lp1 (0, b1;E) for x2 ∈ (0, b2) , genera-ted by problem (2.7). By virtue of [1, Theorem 4.5.2], Lp1 (0, b1;E) ∈ UMD for p1 ∈ (1,∞).Moreover, in view of [10] the operator B is R−positive in Lp1 (0, b1;E). Hence, the pro-blem (2.8) has a unique solution u ∈W [2]p,α(2) (G2;E (A) ;E) for f ∈ Lp (G2;E), |arg λ| ≤ ϕwith sufficiently large |λ| and (2.6) holds for n = 2. By continuing this process we obtainthe assertion. Theorem 2.2. Let the conditions of Theorem 2.1 are satisfied andAk (x)A−(12−ν) (x)∈C (G¯;L (E))for 0 < ν < 12 . Then, problem (2.1) has a unique solution u ∈ W [2]p,α (G;E (A) , E) for f ∈Lp (G;E), |arg λ| ≤ ϕ with sufficiently large |λ| and the coercive uniform estimate holds(2.9)n∑k=12∑i=0|λ|1− i2∥∥∥∥∂[i]u∂xik∥∥∥∥Lp(G;E)+ ‖Au‖Lp(G;E) ≤ C ‖f‖Lp(G;E) .Proof. By second assumption and by [5, Theorem 3.5] for all h > 0 we have the followingEhrling-Nirenberg-Gagliardo type estimate(2.10) ‖L1u‖Lp(G;E) ≤ hµ ‖u‖W [2]p,α(G;E(A),E) + h−(1−µ) ‖u‖Lp(G;E) .Let O denote the operator generated by problem (2.5) andL1u =n∑k=1Ak (x)∂[1]u∂xk.By using the estimate 2.10 we obtain that there is a δ ∈ (0, 1) such that∥∥∥L1 (O + λ)−1∥∥∥B(X)< δ.Hence, from perturbation theory of linear operators we obtain the assertion. Mixed problems for degenerate abstract parabolic equations and applications 2513. ABSTRACT CAUCHY PROBLEM FOR DEGENERATE PARABOLIC EQUATIONConsider the IBVP for degenerate parabolic equation:(3.11)∂u∂t+n∑k=1ak (xk)∂[2]u∂x2k+A (x)u+ du = f (x, t) , t ∈ (0, T ) , x ∈ G,Lkju = 0(3.12) u (x, 0) = 0, t ∈ (0, T ) , x(k) ∈ Gk,where u = u (x, t) is a solution, δki, βki are complex numbers, ak are complex-valuedfunctions on G, A (x) is a linear operator in a Banach space E, domains G, Gk, Gk0, Gkb,σik and x(k) are defined in the section 2.For p = (p0, p) , GT = (0, T ) × G, Lp,γ (GT ;E) will denote the space of all E-valuedweighted p-summable functions with mixed norm.Theorem 3.3. Suppose all conditions of Theorem 2.1 are satisfied for ϕ > pi2 . Then, for f ∈Lp (GT ;E) and sufficiently large d > 0 problem (3.11) - (3.12) has a unique solution belongingto W 1,[2]p,α (GT ;E (A) , E) and the following coercive uniform estimate holds∥∥∥∥∂u∂t∥∥∥∥Lp(GT ;E)+2∑k=1∥∥∥∥∂[2]u∂x2k∥∥∥∥Lp(GT ;E)+ ‖Au‖Lp(GT ;E) ≤ C ‖f‖Lp(GT ;E) .Proof. The problem (3.11) can be expressed as the following abstract Cauchy problem(3.13)dudt+ (O + d)u (t) = f (t) , u (0) = 0.By virtue of [9], O is R-positive in X = Lp (G;E), i.e O is a generator of an analyticsemigroup in X. Then by virtue of [10, Theorem 4.2] problem (3.13) has a unique solutionu ∈ W 1p0 (0, T ;D (O) , X) for f ∈ Lp0 (0, T ;X) and sufficiently large d > 0. Moreover, thefollowing uniform estimate holds∥∥∥∥dudt∥∥∥∥Lp0 (0,T ;X)+ ‖Ou‖Lp0 (0,T ;X) ≤ C ‖f‖Lp0 (0,T ;X) .Since Lp0 (GT ;X) = Lp (GT ;E) , by Theorem 2.1 we have‖(O + d)u‖Lp0 ((0,T );X) = D (O) .Hence, the assertion follows from the above estimate. 4. NONLINEAR DEGENERATE ABSTRACT PARABOLIC PROBLEMIn this section, we consider IBVP for the following nonlinear degenerate parabolicequation(4.14)∂u∂t+n∑k=1ak (xk)∂[2]u∂x2k+B((t, x, u,D[1]u))u = F(t, x, u,D[1]u),(4.15) Lkju = 0, u (x, 0) = 0, t ∈ (0, T ) , x ∈ G, x(k) ∈ Gk,where u = u (x, t) is a solution, αkji, βkji are complex numbers, ak are complex-valuedfunctions on [0, bk]; domains G, Gk, Gk0, Gkb and σik, x(k) are defined in the section 2 andD[i]k u =∂[i]u∂xik=(xαkk∂∂xk)iu (x, t) , 0 ≤ αk < 1.252 Veli. B. Shakhmurov and Aida SahmurovaLet GT = (0, T )×G. Moreover, we letα = α (x) =n∏k=1xαkk .G0 =n∏k=1(0, b0k) , G =n∏k=1(0, bk) , bk ∈ (0, b0k) ,T ∈ (0, T0) , Bki =(W 2,p (Gk, E (A) , E) , Lp (Gk;E))ηik,p,ηik =mki +1p(1−αk)2, B0 =n∏k=11∏i=0Bki.Theorem 4.4. Assume the following hold:(1) E is an UMD space and 0 ≤ α1, α2 < 1− 1p , p ∈ (1,∞);(2) ak are continuous functions on [0, bk] , ak (xk) < 0, for all x ∈ [0, bk] , αkmk1 6= 0,βkmk2 6= 0, k = 1, 2, ..., n;(3) there exist Φki ∈ Bki such that the operator B (t, x,Φ) for Φ = {Φki} ∈ B0 is R-positivein E uniformly with respect to x ∈ G0 and t ∈ [0, T0] ; moreover,B (t, x,Φ)B−1(t0, x0,Φ) ∈ C (G¯;L (E)) , t0 ∈ (0, T ) , x0 ∈ G;(4) A = B(t0, x0,Φ): GT ×B0 → L (E (A) , E) is continuous. Moreover, for each positive rthere is a positive constant L (r) such that∥∥[B (t, x, U)−B (t, x, U¯)] υ∥∥E≤ L (r)∥∥U − U¯∥∥B0‖Aυ‖Efor t ∈ (0, T ) , x ∈ G, U, U¯ ∈ B0, U¯ = {u¯ki} , u¯ki ∈ Bki, ‖U‖B0 ,∥∥U¯∥∥B0≤ r, υ ∈ D (A) ;(5) the function F : GT × B0 → E such that F (., U) is measurable for each U ∈ B0 andF (t, x, .) is continuous for a.a. t ∈ (0, T ), x ∈ G. Moreover, ∥∥F (t, x, U)− F (t, x, U¯)∥∥E≤Ψr (x)∥∥U − U¯∥∥B0for a.a. t ∈ (0, T ), x ∈ G, U, U¯ ∈ B0 and ‖U‖B0 ,∥∥U¯∥∥B0≤ r; f (.) =F (., 0) ∈ Lp (GT ;E).Then there is a T ∈ (0, T0) and a bk ∈ (0, b0k) such that problem (4.14) - (4.15) has a uniquesolution belonging to W 1,[2]p,α (GT ;E (A) , E) .Proof. Consider the following linear problem∂w∂t+n∑k=1ak (xk)∂[2]w∂x2k+ du = f (x, t) , x ∈ G, t ∈ (0, T ) ,(4.16) Lkjw = 0, w (x, 0) = 0, t ∈ (0, T ) , x ∈ G, x(k) ∈ Gk, d > 0.By Theorem 3.3 there is a unique solution w ∈ W 1,[2]p,α (GT ;E (A) , E) of the problem(5.3) for f ∈ Lp (GT ;E) and sufficiently large d > 0 and it satisfies the following coerciveestimate‖w‖W1,[2]p,α (GT ;E(A),E)≤ C0 ‖f‖Lp(GT ;E) ,uniformly with respect to b ∈ (0 , b0], i.e., the constant C0 does not depends on f ∈Lp (GT ;E) and b ∈ (0 b0] whereA (x) = B (x, 0) , f (x) = F (x, 0) , x ∈ (0, b) .We want to solve the problem (4.14) - (4.15) locally by means of maximal regularity of thelinear problem (4.16) via the contraction mapping theorem. For this purpose, let w be asolution of the linear BVP (4.16) Consider a ballBr = {υ ∈ Y, υ − w ∈ Y1, ‖υ − w‖Y ≤ r} .Mixed problems for degenerate abstract parabolic equations and applications 253For given υ ∈ Br, consider the following linearized problem(4.17)∂u∂t+n∑k=1ak (xk)∂[2]u∂x2k+A (x) = F (x, V ) + [B (x, 0)−B (x, V )] υ,(4.18) Lkju = 0, u (x, 0) = 0, t ∈ (0, T ) , x ∈ G, x(k) ∈ Gk.where V = {υki} , υki ∈ Bki. Define a map Q on Br by Qυ = u, where u is solution of(4.17). We want to show that Q (Br) ⊂ Br and that Q is a contraction operator provided Tand bk are sufficiently small, and r is chosen properly. In view of separability propertiesof the problem (4.16) we have‖Qυ − w‖Y = ‖u− w‖Y ≤ C0 {‖F (x, V )− F (x, 0)‖X +‖[B (0,W )−B (x, V )] υ‖X} .By assumption (4) we have‖[B (0,W ) υ −B (x, V )] υ‖X ≤ supx∈[0,b]{‖[B (0,W )−B (x,W )] υ‖L(E0,E)+ ‖B (x,W )−B (x, V )‖L(E0,E) ‖υ‖Y}≤[δ (b) + L (R) ‖W − V ‖∞,E0][‖υ − w‖Y + ‖w‖Y ] ≤{δ (b) + L (R) [C1 ‖υ − w‖Y + ‖υ − w‖Y ][‖υ − w‖Y + ‖w‖Y ]} ≤ δ (b) + L (R) [C1r + r] [r + ‖w‖Y ] ,whereδ (b) = supx∈[0,b]‖[B (0,W )−B (x,W )]‖B(E0,E) .In view of above estimates, by suitable choice of µR, LR and for sufficiently small T ∈(0, T0) and bk ∈ (0 , b0k] we have‖Qυ − w‖Y ≤ r,i.e.Q (Br) ⊂ Br.Moreover, in a similar way we obtain‖Qυ −Qυ¯‖Y ≤ C0 {µRC1 +Ma + L (R) [‖υ − w‖Y + C1r] +L (R)C1 [r + ‖w‖Y ] ‖υ − υ¯‖Y }+ δ (b) .By suitable choice of µR, LR and for sufficiently small T ∈ (0, T0) and bk ∈ (0, b0k) weobtain ‖Qυ −Qυ¯‖Y < η ‖υ − υ¯‖Y , η < 1, i.e. Q is a contraction operator. Eventually, thecontraction mapping principle implies a unique fixed point ofQ inBr which is the uniquestrong solution u ∈W 1,[2]p,α (GT ;E (A) , E) . 254 Veli. B. Shakhmurov and Aida SahmurovaREFERENCES[1] Amann, H., Linear and quasilinear parabolic problems, 1, 2, Birkhauser, Basel, 1995[2] Agarwal, R., O’Regan, D., and Shakhmurov, V. B., Separable anisotropic differential operators in weighted ab-stract spaces and applications, J. Math. Anal. Appl., 338 (2008), 970–983[3] Ashyralyev, A., Claudio, C. and Piskarev, S., On well-posedness of difference schemes for abstract elliptic problemsin Lp spaces, Numer. Func. Anal. Optim., 29 (2008), 43–65[4] Favini, A. and Yagi, A., Degenerate Differential Equations in Banach Spaces, Taylor & Francis, Dekker, New-York, 1999[5] Shakhmurov, V., Embedding theorems and maximal regular differential operator equations in Banach-valuedfunction spaces, J. Inequal. Appl., (4) (2005), 605–620[6] Shakhmurov, V. 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Ordinary and Partial Differential Equations, Chap-man and Hall /CRC, Boca Raton, 20001OKAN UNIVERSITYMECHANICAL ENGINEERINGAKFIRAT, TUZLA 34959 ISTANBUL, TURKEY2KHAZAR UNIVERSITYMATHEMATICSBAKU, AZERBAIJANE-mail address: veli.sahmurov@okan.edu.tr3OKAN UNIVERSITYENVIRONMENTAL ENGINEERINGAKFIRAT, TUZLA 34959 ISTANBUL, TURKEYE-mail address: aida.sahmurova@okan.edu.tr

Mixed problems for degenerate abstract parabolic equations and applications

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Mixed problems for degenerate abstract parabolic equations and applications

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