CMAT Centro de Matemática da Universidade do Minho

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1 Universidade do Minho CMAT Centro de Matemática da Universidade do Minho Campus de Gualtar Braga Portugal wwwcmatuminhopt Universidade do Minho Escola de Ciências Centro de Matemática Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Stanislav Antontsev a,b Fernando Miranda c Lisa Santos b a CMAF-CIO, Universidade de Lisboa, Portugal b Novosibirsk State University, Russia c CMAT and Departamento de Matemática e Aplicações, Universidade do Minho, Portugal Information Keywords: Electromagnetic problems px, t-curl systems Variable exponents Blow-up Extinction in time Original publication: J Math Anal Appl 44, 3-3, 16 DOI: 1116/jjmaa16345 wwwelseviercom Abstract We study a class of px, t-curl systems arising in electromagnetism, with a nonlinear source term Denoting by h the magnetic field, the source term considered is of the form λh h σ where λ 1,, 1: when λ 1, we consider < σ and for λ = 1 we have σ 1 We introduce a suitable functional framework and a convenient basis that allow us to apply the Galerkin s method and prove existence of local or global solutions, depending on the values of λ and σ We study the finite time extinction or the stabilization towards zero of the solutions when λ 1, and the blow-up of local solutions when λ = 1 1 Introduction The study of partial differential equations or systems with variable exponents is a recent research topic which had a very quick development that started when it was understood that variable exponents give better descriptions of the behavior of certain materials or phenomena To the authors knowledge, this is one of the first works involving the px, t-curl operator In this work we intend to generalize the results in [1] to a similar problem but now with variable exponents Let be a bounded simply connected domain of R 3 with a C 1,1 boundary denoted by Γ, T R +, Q T =, T and Σ T = Γ, T In what follows, vector functions and spaces of vector functions will be denoted by boldface symbols We will use x to denote the partial derivative of a function with respect to the variable x The divergence of a vector function h = h 1, h, h 3 is denoted by h = x1 h 1 + x h + x3 h 3 and the curl of h by h = x h 3 x3 h, x3 h 1 x1 h 3, x1 h x h 1

2 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism We recall the identity h = h h, 1 where h = h 1, h, h 3 and h i = h i, i = 1,, 3 We wish to prove existence of solution for the system t h + h px,t h = fh, h = in Q T, a h px,t h n =, h n = on Σ T, b h, = h in, c where fh = λ h h σ with λ 1,, 1 and σ a positive constant Using the Galerkin s method, we prove existence of solution h XQ T H 1, T ; L to the above problem The space XQ T = v L Q T : v L p, Q T, v =, v n Γ = is the suitable functional framework to solve weakly the system A first difficulty is the characterization of the space W p = v L p : v L p, v =, v n Γ =, as a subspace of the Orlicz-Sobolev space W 1,p where the seminorm L p is a norm equivalent to the one induced by the W 1,p -norm see Theorem 1 Others difficulties are the definitions of suitable countable topological bases of W p and of XQ T see Proposition 3 and Proposition 3, respectively We prove existence of solutions to this system imposing that, besides natural assumptions, the function p, is decreasing in time We study the finite time extinction or blow-up of the solutions, depending on the values of the parameters λ and σ Functional framework spatial variables The main purposes of this section are the characterization of the space of divergence free vector functions belonging to L p with curl in L p and normal null trace as well as the construction of a suitable topological basis for this space 1 The Orlicz Sobolev spaces L p and W 1,p We start by collecting a few well known facts from the theory of Sobolev spaces with variable exponent For details about this theory and an exhaustive review of the existing bibliography, see the monograph [5] Let p : [1, be a continuous function We will use the notation p C log if p satisfies ζ 1, ζ, ζ 1 ζ < 1, pζ 1 pζ ω ζ 1 ζ, lim sup τ + ωτ log 1 τ = C 3 Defining we denote A p f = f p, L p = f : R : f is measurable and A p f < The space L p, equipped with the Luxemburg norm is a Banach space f L p = inf λ > : A p f λ 1,

3 S Antontsev, F Miranda, L Santos 3 From now on we assume that p = min x px and p + = max px, 1 < p, p + < 4 x We define W 1, p = u L p : u p L 1, and we endow it with the norm We consider also with the norm u W 1,p = u L p + u L p 5 1, p W = u W 1, p : u Γ =, u W 1,p = u L p Details about the trace of a function u W 1,p can be found in [5] Let us indicate the basic properties of the spaces L p, W 1,p and W 1,p that we will need in the rest of this paper: 1 From the definition of the norm in L p, we conclude that min f p L p, f p+ A L p p f max f p L p, ; 6 f p+ L p Hölder s inequality is verified, ie, for all f L p, g L p with px 1,, p x = px px 1, the following inequality holds: 1 f g p + 1 p f L p g L p f L p g L p ; 3 We also have that for q p L p L q and f L q C f L p ; 4 The space W 1,p is separable and reflexive, provided that p C and satisfies 4; 5 Assuming 3, we have D is dense in W 1,p and this last space can be defined as the completion of D with respect to the norm 5 The density of smooth functions in the space W 1,p is crucial for the understanding of these spaces The condition of log-continuity of p is the best known and the most frequently used sufficient condition for the density of D in W 1,p see [4, 5] Although this condition is not necessary and can be substituted by other conditions see [5, Chapter 9] for a discussion of this question we keep it throughout the paper for the sake of simplicity of presentation; 6 Observing that W 1,p W 1,p, the Sobolev inequality f Lq C f W 1,p holds, with 1 q < 3p positive constant 3 p if p < 3, any q if p = 3 and q = if p > 3 Here C = Cp, is a

4 4 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism The space W p We define W p = v L p : v L p, v =, v n Γ =, endowed with the norm v W p = v L p + v L p Theorem 1 Assume that 1 < p p p + < and p satisfies 3 Then W p is a closed subspace of W 1,p n, where W 1,p n = v W 1,p : v n Γ = Besides, if p > 6 5, then L p is a norm in W p equivalent to the norm induced from W 1,p In particular, v W 1,p C v L p, where C = Cp, p +, Proof The inclusion v W 1,p n : v = W p is immediate We need to prove that W p W 1,p n Let u L p be such that u L p, u = and u n Γ = Since Γ is compact, we can find z 1,, z k Γ and r 1,, r k R + such that, denoting B i = Bz i, r i the ball centered in z i with radius r i and setting B i = Bz i, ri, we have: 1 k i=1 B i Γ; For i = 1 k, there exists a set V + i R R + and a smooth function Φ i which is a bijection from B i to V + i and such that ΦΓ B i = V + i R ; 3 Denote, for i fixed, y = Φ i x and y i = Φ i z i Let L = a jk yjy k + b j yj be the expression of the Laplacian in the new variables for details see [6, p 97] We choose the radius r i of the ball B i sufficiently small such that: a max a jky a jk y i < 1 y Φ ib i 36 C, 7 b where C = C p, p +, is the constant for the W 1,p estimate satisfied by the solutions of the equations a jk y i y jy k v = g in, where g is any element of W p ; v = on Γ, diamv + i b j W 1, 1 4 C, 8 where diamv + i represents the diameter of V + i defined in item, b j are the coefficients of the first order terms of the operator defined above in this item and C is the constant mentioned in a; 4 Since is compact, we can find z k+1,, z m and r k+1,, r m R + such that m B i i=1

5 S Antontsev, F Miranda, L Santos 5 For i k + 1,, m, let η i DB i be such that η i 1 and η i Bi 1 The function v i = η i u satisfies the following problem v i = η i u η i u u η i in B i 9 v i = on B i Observe that F i u = η i u η i u u η i W 1,p B i and so v i W 1,p B i see p 439 of the proof of [5, Theorem 141] Then v i W 1,p B i and u W 1,p B i v i W 1,p B i C F iu W 1,p B i, where C = C p, p +, > In what follows C represents different positive constants Using the identity 1 and recalling that u =, we have η i u W 1,p B i η i L B i sup ϕ W 1,p B i ϕ W 1,p B i 1 η i L B i sup ϕ W 1,p B i ϕ W 1,p B i 1 u, ϕ W 1,p B i W 1,p B i B i u ϕ η i L B i C u L p B i, sup ϕ W 1,p B i ϕ W 1,p B i 1 u L p B i ϕ L p B i η i u W 1,p B i η i L B i sup ϕ W 1,p B i ϕ W 1,p B i 1 η i L B i sup ϕ W 1,p B i ϕ W 1,p B i 1 u, ϕ W 1,p B i W 1,p B i B i u ϕ η i L B i sup ϕ W 1,p B i ϕ W 1,p B i 1 u L p B i ϕ W 1,p B i C u L p B i and So, u η i W 1,p B i η i L B i sup ϕ W 1,p B i ϕ W 1,p B i 1 η i L B i C u L p B i sup ϕ W 1,p B i ϕ W 1,p B i 1 B i u ϕ F i u W 1,p B i C u L p B i + u L p B i u L p B i ϕ L p B i

6 6 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Let now i 1,, k and V + i Given a function ϕ : V + i be the sets as defined in item Define V i = y 1, y, y 3 : y 1, y, y 3 V + i and V i = V + i V i R, we define ϕ : V i R by even reflection, ie, ϕy 1, y, y 3 if y 1, y, y 3 V + i ϕy 1, y, y 3 = ϕy 1, y, y 3 if y 1, y, y 3 V i For a given function v : B i R 3 we call v = v Φ 1 i, where Φ i is the change of variables defined in item Recalling that y = y 1, y, y 3 = Φ i x, we have vy 1, y, y 3 = vx 1, x, x 3 Let η i DB i, η i 1, η i Bi 1 For v i = η i ū we have v i x = a jk y y iy k v i y + b j y yj v i y, where a jk j,k is strictly positively defined and a jk, b j C V i for details see [6, p 97] Then the problem, in the variable x, v i x = F i ux in B i v i x = on B i v i x n = on Γ B i is transformed into the problem, in the variable y, a jk y y iy k v i y = G i uy + b j y yj v i y in V i, 1 v i y = on V i where G i u is the function that corresponds to F i u defined in 9, but in the new variable y Recalling the definition of y i = Φ i z i given in item 3, we can rewrite the problem 1 as follows a jk y i y iy k v i y = G i uy + b j y yj v i y a jk y i a jk y y iy k v i y in V i, v i y = on V i and so v i W 1,p V i C G i u W p V i + b jy yj v i y W p V i We are going to estimate first the terms b j y yj v i y W p V i : + a jk y i a jk y y iy k v i W p V i 11 b j yj v i, ϕ W 1,p V i W 1,p V = v i i, yj b j ϕ + b j yj ϕ W 1,p V i W 1,p V i v i ϕ + yj ϕ b j W 1, V i V i b j W 1, V i v i L p V ϕ i W 1,p V i Applying the Poincaré inequality see [5, Theorem 84] we obtain b j v i, ϕ W 1,p V i W 1,p V i diamv i b j W 1, V i v i W 1,p V i ϕ W i,p V i and so, recalling that diamv i diamv + i and 8, b j yj v i W 1,p V i = sup ϕ W 1,p V i ϕ W 1,p V i 1 b j v i, ϕ W 1,p V i W 1,p V i 1 1 C v i W 1,p V i 1

7 S Antontsev, F Miranda, L Santos 7 Returning to 11, using assumption 7 and estimate 1, v i W 1,p V i C G i u W p V i + b j yj v i W 1,p V i + a jk y i a jk y L V i sup ϕ W 1,p V i ϕ W 1,p V i 1 yk v i yi ϕ V i C G i u W p V i + 1 v i W 1,p V i and we conclude that So v i W 1,p V i C G iu W p V i u i W 1,p B i v i W 1,p B i C 1 v i W 1,p V i C G i u W p V i C 3 F i u W p B i C 4 u L p B i + u L p B i To conclude that u W 1,p it is enough to use a partition of unity subordinated to the covering B i i=1,,m Applying Peetre s Lemma, we conclude that there exists a positive C = Cp, p +, such that v W p v W 1,p n C v L p For details of the proof of this inequality, when p is constant, see [7, Theorem 1] Remark Let v W p From item 6 of Subsection 1 and the previous theorem, the inequality holds, with 1 q < positive constant 3p 3 p 3 A basis for W p v L q C v L p 13 if p < 3, any q if p = 3 and q = if p > 3 Here C = C p, p +, is a We wish to find out an appropriate countable topological basis of W p, to be able to define a family of approximating problems in finite dimensional subspaces Proposition 3 There exists a countable topological basis ψ n n of W p such that, for all n N, ψ n X,where X = v C 1 : v =, v n Γ = Proof A function u W p belongs to W 1,p So, there exists an extension, still denoted by u, belonging to W 1,p R 3 see [5, Theorem 85] Given ε >, let ρ ε DR 3 be a mollifier, define u ε = u ρ ε = u 1 ρ ε, u ρ ε, u 3 ρ ε and recall that u ε DR 3 and u ε u in W 1,p R 3 ε Let v ε be a solution of the problem v ε = u ε in v ε n = uε n on Γ

8 8 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Then v ε W,q, for any 1 < q < and, in particular, v ε C 1 Setting z ε = u ε v ε, we observe that z ε =, z ε C L p and z ε n Γ = So, z ε X W p Besides, z ε u W p = z ε u L p = u ε u L p u ε u L p R 3 ε Since W p is a separable space, it admits a countable topological basis ϕ n n Given n, m N, we construct a function z n,m as above, with ε = 1 m, such that z n,m X and z n,m ϕ n W p 1 m But this implies that z n,m n,m N is a countable subset of X such that k l µ ij z i,j : k, l N, µ ij R for i = 1,, k and j = 1,, l i=1 j=1 is dense in W p So, we can extract from z n,m n,m a topological countable vector basis of W p 3 Functional framework and weak formulation In this section we present an adequate functional framework to define a weak formulation of problem to which we will prove existence of solution Let p : QT 1, be a given function We will use the notation p C log Q T if p satisfies the log-continuity condition in the cylinder Q T : ζ 1, ζ Q T, ζ 1 = x, t, ζ = y, τ, ζ 1 ζ = x y + t τ < 1, Throughout the rest of the text we use the notations Let endowed with the norm XQ T = p = min ζ Q T pζ, pζ 1 pζ ω ζ 1 ζ, p + = max ζ Q T pζ v L Q T : v L p, Q T, v =, v n Γ = v XQT = v L Q T + v L p, Q T Remark 31 Recall that, by Theorem 1, given v XQ T, we have, for ae t, T, v, t p,t C 1 v, t p,t, lim sup ωs log 1 = C 14 s s + where, setting p + t = sup px, t and p t = inf px, t, C 1 is a positive constant depending only on p + t, x x p t and Noticing that p p t px, t p + t p +, the constant C 1 can be chosen independent of t Proposition 3 Let ψ n n be a basis of W p defined as in Proposition 3 Then the set m V = ζ k tψ k x : ζ k C 1 [, T ], m N is dense in XQ T k=1

9 S Antontsev, F Miranda, L Santos 9 Proof We know that the set m Z = µ k ψ k : µ k R, m N is dense in W p On the other hand, the set Y Q T = αtψx : α C 1 [, T ], ψ W p k=1 is dense in L q, T ; W p, for any 1 < q < But, as the conclusion follows XQ T L min,p, T ; W p, We can now present a weak formulation of problem : to find h XQ T H 1, T ; L such that, for ae t, T, t ht ψ + ht p,t ht ψ = fht ψ, ψ W p, 15a h, = h 15b Observe that, when h XQ T H, T ; L then h C [, T ]; L and so h, has a meaning 4 The case λ 1, and < σ σ In this section we prove existence of global solutions of the problem 15, when fh = λh h λ 1, We also study the finite time extinction or asymptotic vanishing in time of the solutions 41 Existence of solution Let ψ n n be a basis of W p defined as in Proposition 3 Assume that h W 1,p,, 16 and let h m be an approximation, in W 1,p,, of h such that h m ψ 1,, ψ m Setting h m t = m ζi m tψ i, then the system of ODE s in the unknowns ζ1 m,, ζm, m t h m t ψ i + h m t p,t h m t ψ i λ h m t ψ i h m t σ = h m, = h m has a solution ζ1 m,, ζm m C 1 [, T ] m The above system is equivalent to t h m t ψ + h m t p,t h m t ψ i=1 σ λ h m t ψ h m t =, ψ ψ 1, ψ m 17a h m, = h m 17b for

10 1 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Proposition 41 Assume that 1 < p p, p + <, p and h satisfy, respectively, 14 and 16 Let h m be a solution of problem 17 Then there exists a positive constant, independent of m, such that h m L,T ;L C, h m L p, Q T C 18 Proof Observe that we can use h m t as a test function in 17, obtaining σ t h m t h m t + h m t p,t λ h m t = Integrating the above equality between and t and denoting Q t =, t, we get and so 1 h m t + h m τ p,τ dτ λ Q t which immediately yields the conclusion t σ h m τ 1 dτ = sup h m t + h m p, h m, t T Q T h m We will now prove that, under stronger assumptions on the function p,, the partial derivative of h with respect to t belongs to L Q T Proposition 4 Assume that 1 < p p, p + <, p and h satisfy, respectively, 14 and 16, and also that there exists a positive constant c such that c t p ae in Q T Then Q T t h m t + sup t T In particular, h m t p,t λ h m t σ p, t σ h m p, λ p, σ Proof We may use t h m as test function in 17, obtaining t h m t + h m t p,t h m t t h m t λ and so where t h m t + d dt I = h m p,t p, t h m σ + c T p 19 t h m L Q T C h m t p,t λ d h m t σ p, t σ dt h m t t h m t h m t σ = = I, p, t log h m t t p, t We evaluate I by the following way: h m t p,t I = 1 p, t log hm p, t t t p, t = h mt p,t 1 p, t log hm 1 p,t log h mt p, t t t p, t + h mt p,t 1 p, t log hm 1<p,t log h mt p, t t t p, t h mt p,t 1 p, t log hm 1 p,t log h mt p, t t t p, t

11 S Antontsev, F Miranda, L Santos 11 e 1 p,, Next we use the following properties of the function F η = ηp, F = F to conclude that p, 1 p, log η, defined for η e 1 p, =, F η = η p, 1 log η, max F η = F 1 = 1 η e p, 1 p,, I c p 3 Integrating equality 1 between and t and using inequality 3, we prove 19 To prove it is enough to notice that, from inequality 6, h m p, 1 p, p max h m p p,, h m p+ p, Theorem 43 Assume that 6 5 < p p, p + <, p and h satisfy 14 and 16, respectively, and also that there exists a positive constant c such that c t p ae in Q T Then problem 15 has a solution h XQ T H 1, T ; L Besides, if σ 1, the solution is unique Proof By the estimates 18 and, there exist G and h such that, at least for a subsequence, we have h m m G t h m m th in Lp, Q T -weak, in L Q T -weak Let q = min, p Recall that, by Remark 31, given v XQ T, we have v, t p,t C 1 v, t p,t, where C 1 is a constant that can be chosen independently of t Then, there exists a positive constant C such that v L q Q T + v L q Q T C v L Q T + v L p, Q T Then h m m is bounded in W 1,q Q T and this space is compactly included in L q Q T Here q is the critical Sobolev exponent and it is greater than because p > 6 5 So, at least for a subsequence, we have h m h strongly in L q Q T and h m x, t hx, t for ae x, t Q T m m Moreover, observing that d dt h, t h m, t = we conclude that h, t h m, t h h m Q T h, t hm, t t h, t t h m, t, 1 QT t h t h m 1 + which proves the strong convergence of h m m to h in L, T ; L N For N N, let ϕt = d k tψ k According to 17 we have k=1 h h m m, t h m t ϕt + h m, t p,t h m, t ϕt λ h m, t ϕt h m, t σ = 4

12 1 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Integrating the last equality with respect to t and passing to the limit as m for fixed N, we obtain t h ϕ + G ϕ λ h ϕ h σ =, 5 Q T Q T Q T first for ϕ = N d k tψ k and after, by density, for any ϕ XQ T k=1 Let Av = v p, v and recall that We will now prove that Q T G ϕ = lim m Q T Av Aw v w 6 Ah m ϕ = Ah ϕ Q T Q T Subtracting 5 with ϕ = h from 4, integrated in the interval [, T ], with ϕ = h m, we derive J m + h m p, G h =, 7 where J m = t h m h m t h h λ h m Q T Q T Q T σ h Q T σ m Subtracting 7 from 6 with w = h m and passing to the limit as m, we arrive at inequality G Av h v 8 Q T Choosing v = h νw, where ν is a real positive number and w is any function in XQ T, and substituting it into 8, we have G Ah νw w Q T Letting ν we obtain the inequality G Ah w Q T and so G = Ah Recalling that h H 1, T ; L then h C [, T ]; L Setting q = minp,, the set h m m is bounded in Z = v L q, T ; W p, t v L Q T and we will prove below that Z is compactly included in C [, T ]; L Observe that q t+δ h m t + δ h m t q = t h m τ dτ t t+δ δ q 1 t h m τ q dτ t δ q 1 t h m q L q Q T δq C We have W p L L q, being the first inclusion compact because p > 6 5 On the other hand h m m is a bounded subset of L q, T ; W p and, denoting τ δ ft = ft + δ, we have τ δ h m h m L,T δ;l q = sup t [,T δ] h m t + δ h m t q δ

13 S Antontsev, F Miranda, L Santos 13 Then, by [8, Theorem 5], h m m is compactly included in C [, T ]; L So, at least for a subsequence, we have h m h in C [, T ]; L and, in particular, h = lim h m = lim h m = h m m m This concludes the proof that h solves the problem 15 To prove the uniqueness of solution in the case σ 1, we follow the steps in [1] The proof in the case λ = is immediate Let λ = 1 and h 1 and h be two solutions of 15 Use h 1 h as test function in the problem solved by h 1 and by h Then we get, after subtraction, 1 h 1 t h t + h 1 p, h 1 h p, h h 1 h + t Q t h 1 σ h 1 h 1 h t h σ h h 1 h = Recalling 6, we get 1 t h 1 t h t + h 1 σ t h 1 h 1 h h σ h h 1 h Calling y 1 t = h 1 t and y t = h t, we have, for σ 1, t t y σ h 1 σ h 1 h 1 h 1 + y σ y σ 1 1 y 1 y σ 1 y 1 1 = t t y σ 1 1 y σ 1 h σ h h 1 h y 1 1 y 1, and the above inequality implies that y 1 t = y t = for ae t, T and t h 1 σ t h 1 h 1 h h σ h h 1 h = Consequently, which implies that h 1 = h ae in Q T 1 h 1 t h t, 4 Finite time extinction and asymptotic behavior We are going to study now the finite time extinction or stabilization in time, towards zero, of the solutions of problem 15, depending on the choice of the parameters λ and σ Theorem 44 Let h be a solution of problem 15 with 1 < p, <, λ = 1, < σ < and h L Then there exists a positive t = 1 σ h σ L such that, for t t, we have ht L = Proof Observing that a solution h of problem 15 satisfies 1 d ht + ht p,t + dt denoting Y t = we obtain that Y satisfies the differential inequality and so ht = ht L Y t + Y t σ ht L = t 1 σ ht σ h σ =, 9

14 14 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Remark 45 Notice that, according to 9, we derive, for the limit case σ = and also for σ >, the asymptotic extinction of the solution when t In fact, using the above notation, if σ =, if σ >, we have Y t Y t Y e t ; Y σ 1 + tσ Y σ Now we are going to investigate the asymptotic behavior of ht with respect to t, where h solves problem 15 when the term fh is absent, ie, λ = Theorem 46 Let h be a solution of problem 15 with 1 < p, p + <, λ = and h L Then there exists a positive t < such that ht L = for t t Proof Taking into account the first estimate in 18 we can assume, without loss of generality, that ht L 1 Using the energy relation 1 d ht + ht p,t =, dt 6 and 13, we can write, for any fixed t, 1 ht L C max ht p,t p 1, ht p,t p + and so ht p+ L = min ht p L, C ht p,t ht p+ L Setting Y t = ht, the last inequality and the energy relation lead us to the following ordinary L differential inequality which gives Y t + CY t p+, 3 Y t p+ Y p+ p+ Ct This completes the proof with t = C p + h p+ L Now we consider a limit situation when 1 < p, sup px, t = p + t and p + t as t 31 x Theorem 47 Let h be a solution of problem 15 with p, satisfying 31, λ = and h L Assume that the exponent p + t is monotone increasing and dt e t p+ t Then there exists a positive t < such that ht L = for t t The proof of this theorem will use the following lemma < 3 Lemma 48 [4, Lemma 67] and [, Lemma 91] Let a nonnegative function Θt satisfy the conditions Θ t + CΘ µt t for ae t with µt, 1 and C a positive constant Θt Θ <, Θ > If the exponent µt is monotone increasing, then Θt for all t t with t defined from the equality t C Θ µs 1 dz ds = e z1 µz

15 S Antontsev, F Miranda, L Santos 15 We present now the proof of Theorem 47 Proof of Theorem 47 Applying Lemma 48 to the inequality 3 we derive t C h p+ s dt ds = L e < t p+ t It follows that there exists a positive t < such that ht L = for t t Remark 49 A simple example of an exponent p + t = sup px, t satisfying the conditions of the Theorem 47 x is p + t = 5 The case λ = 1 and σ 1 1 α log t t, 1 < α, e t σ In this section we prove existence of global or local solutions of the problem 15, for fh = h h and σ 1 We also study the blow-up of local solutions 51 Existence of solution We consider, as in Subsection 41, a basis ψ n n of W p, defined as in Proposition 3 Assuming that h satisfies 16 let h m be an approximation, in W 1,p,, of h such that h m ψ 1,, ψ m Denoting m h m t = ζi m tψ i, i=1 then the system of ODE s in the unknowns ζ m 1,, ζ m m, t h m t ψ i + h m t p,t h m t ψ i = h m t ψ i h m t σ h m, = h m has a solution ζ1 m,, ζm m C 1 [, T ] m This problem is equivalent to the following problem t h m t ψ + h m t p,t h m t ψ = h m t ψ i h m, = h m h m t σ, ψ ψ 1, ψ m 33a Theorem 51 Assume that 6 5 < p p, p + <, p and h satisfy 14 and 16, respectively Assume, in addition, that there exists a positive constant c such that c t p ae in Q T 1 If If 33b 1 σ max, p 34 then problem 15 has a solution h XQ T H 1, T ; L for any positive T σ > max, p + 35 then problem 15 has a solution h XQ T H 1, T ; L for a small T max >

16 16 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Proof First of all we derive a priori estimates, independent of m, for the approximations h m These estimates will be global, for any finite T, in 34 and local, for a small T max >, in 35 We start by proving that, in item 1, for any finite T there exists a positive constant C such that h m L,T ;L + h m p, C, 36 Q T t h m + sup h m t p,t C 37 Q T t T Using h m t as test function in 33, we obtain σ 1 d h m t + h m t p,t = h m t 38 dt We split now the proof of the estimate 36 in two cases: 1 σ and < σ < p The case 1 σ The function Y t = and h m t satisfies the inequality Y t Y t σ So h m t σ σt + h m t e T and the estimate 36 is immediately obtained The case < σ < p h m t σ, if σ <, h m t, if σ =, According to the formula 13 of Remark and the inequalities 6, for any fixed t [, T ], we have that σ h m t C hm t σ L p,t σ C h m t p,t p, if h m t p,t > 1 C σ h m t p,t p +, if h m t p,t 1 Taking into account that σ < p p + and applying Young inequality, we obtain σ h m t 1 h m t p,t + C Substituting the last estimate in 38 and integrating with respect to t, we conclude the proof of the estimate 36 Now we derive the estimate 37 Using t h m as test function in 33 we obtain t h m t + d h m t p,t = I + d σ 1 h m t dt p, t dt σ where I is defined in and can be estimated, as in 3, by I we have Q t t h m + h m t p,t p, t h m p, + c p, p T + 1 σ, c p Integrating between and t, σ h m t 1 σ σ h m 39

17 S Antontsev, F Miranda, L Santos 17 Applying the estimate in 36 for h m L,T ;L, we obtain estimate 37 Now we derive local estimates 36 and 37, for T > small enough, in the case of item Assuming that 1 t < T max = σ h σ, L it follows from 38 that h m t 1 tσ σ h m σ h m and we obtain the estimate 36 Substituting the last estimate in 39 we obtain 37 for t < T max To conclude the proof of the existence of global solution, in the case 34, and local solution, in the case 35, we only need to argue as in the last part of the proof of Theorem 43 5 Blow-up In this subsection, following some ideas of paper [3], we study the blow-up of local solutions of problem 15 We consider first the case where p depends only on x and afterwards on x, t Theorem 5 Let h be a solution of problem 15 with 1 < p <, λ = 1 and h L Suppose that h p E = 1 σ h p σ and σ > max, p + Then, if µ 1 σ, 1 p +, the solutions of problem 15 blow up on the interval, t max, where Proof Let t max = Et = µσ σ µσ 1 h σ L ht p 1 σ ht p σ Using t ht as test function in 15, we obtain t ht ht p + t = 1 p σ t and so ie Setting we have t h + Q t ht p = 1 p σ F t = 1 ht σ 1 σ ht σ h σ h p + p Et + t h = E Q t 4 F t = 1 Q t h, ht and F t = Using now ht as test function in 15, we get F t + ht p = t ht ht ht σ

18 18 Blow-up and finite time extinction for px, t-curl systems arising in electromagnetism Recalling the definition of Et and 4 we have, for any µ, µf t Et+µ ht σ 1 ht p p + µ ht p + µ 1 ht σ σ So, for 1 σ < µ < 1 p +, and consequently for concluding the proof ht µf t µ 1 σ t < t max = σ F t σ µσ 1 µσ σt + h σ L 1 σ µσ σ µσ 1 h σ, 41 L Theorem 53 Let h be a solution of problem 15 with 1 < p, <, c t p ae in Q T, where c is a positive constant, λ = 1 and h L Suppose that E = h p, 1 σ h <, p, σ σ > max, p + and c small enough such that t max < E p, for t max defined in 41 Then any c solution of problem 15 blows up when t t max Proof Using t ht as test function in 15 and recalling that p depends now on x, t, following the calculations done in the proof of Proposition 4, we obtain where t ht + t I = Integrating between and t, we have t h + Q t Setting we conclude that ht p,t ht p,t = p, t Et = p, t t ht p,t = I + 1 p, t σ t ht σ, 1 + p, t log ht t p, t h p, I ht σ 1 h σ p, σ σ ht p,t 1 p, t σ Et + t h = E + Q t t ht σ, Iτdτ But, from 3, we know that I c p So, choosing t small enough, we have t t E + and we conclude the proof as in the previous theorem t I

19 S Antontsev, F Miranda, L Santos 19 Acknowledgment The research was partially supported by the Research Center CMAF-CIO of the University of Lisbon, Portugal, by the Research Center CMAT of the University of Minho, Portugal, with the Portuguese Funds from the Fundação para a Ciência e a Tecnologia, through the Projects UID/MAT/4561/13 and PEstOE/MAT/UI13/14, respectively, and by the Grant No of Russian Science Foundation, Russia References [1] S Antontsev, F Miranda, L Santos, A class of electromagnetic p-curl systems: blow-up and finite time extinction, Nonlinear Anal doi:1116/jna111 [] S N Antontsev, S Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J Math Anal Appl doi:1116/jjmaa9719 [3] S Antontsev, S Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J Comput Appl Math doi:1116/jcam116 [4] S Antontsev, S Shmarev, Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blow-up, Atlantis Studies in Differential Equations, Atlantis Press, 15 [5] L Diening, P Harjulehto, P Hästö, M Růžička, Lebesgue and Sobolev spaces with variable exponents, Vol 17 of Lecture Notes in Mathematics, Springer, Heidelberg, 11 doi:117/ [6] D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 1, reprint of the 1998 edition [7] F Miranda, J F Rodrigues, L Santos, A class of stationary nonlinear Maxwell systems, Math Models Methods Appl Sci doi:1114/s [8] J Simon, Compact sets in the space L p, T ; B, Ann Mat Pura Appl