Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent

Boundary Value Problems, Apr 2014

The authors investigate a degenerate parabolic equation with delay and nonlocal term, which describes slow diffusive processes in physics or biology. The existence of a nonnegative nontrivial periodic solution is obtained through the use of the Leray-Schauder degree method. MSC: 35D05, 35K55.

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Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent

Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent Zhongqing Li 0 Wenjie Gao 0 0 College of Mathematics, Jilin University , Changchun, 130012 , PR China The authors investigate a degenerate parabolic equation with delay and nonlocal term, which describes slow diffusive processes in physics or biology. The existence of a nonnegative nontrivial periodic solution is obtained through the use of the Leray-Schauder degree method. MSC: Primary 35D05; secondary 35K55 degenerate parabolic equation; periodic solution; variable exponent; topological degree; De Giorgi iteration 1 Introduction (x, t) ∈ T , x ∈ . employed for some technological applications, such as medical rehabilitation equipment and shock wave absorber. When p(x) is a constant and m > , p > , the model describes the slow diffusion process in physics, which has been extensively investigated; see [–]. For example, in [], the authors studied the following doubly degenerate parabolic equation with logistic periodic sources: ∂u – div ∇um p–∇um = uα a – buβ . ∂t ∇ σ um + u  + η 2 Preliminaries and the regularized problems of (1.1) u(x) p(x) dx < ∞ , equipped with the following Luxemburg norm: |u|Lp(x)( ) := inf λ >  : dx ≤  . ≤ |u|Lp(x)( ) ≤ max uv dx ≤ i.e. the embedding Lp(x)( ) → Lp(x)( ) is continuous. If q ∈ C( ) and  ≤ q(x) < p∗(x), for any x ∈ Lq(x)( ) is continuous and compact. Here , then the embedding W,p(x)( ) → p∗(x) := ⎩ +∞, p(x) ≥ N . We next define the weak solutions to problem (.).  = – auϕ + uϕ and ϕ(·, t)|∂ =  for t ∈ [, T ]. As in [], we introduce the following regularized problem: ⎧⎪ ∂∂ut – div{(|∇(σ um + u)| + η) p(x)– ∇(σ um + u)} ⎪⎪⎪⎨⎪ = [a – K (ξ , t)u(ξ , t – τ ) dξ ]u, ⎪⎪ u(·, t)|∂ = , ⎪⎪⎪⎩ u(·, ) = u(·, T ),  = ∇ σ umη + u η  + η and ϕ(·, t)|∂ =  for t ∈ [, T ]. (σ , f ) → u η = G η(σ , f ), a.e. t ∈ (, T ), 3 A priori estimates to the regularized problem First of all, the following modified De Giorgi iteration lemma will be useful (we give a proof in the Appendix). Lemma . (Iteration lemma) Suppose ϕ(t) is a nonnegative and nonincreasing function on [K, +∞), it satisfies ϕ(h) ≤ ϕ(k + d) = , ≤ Ku, mq Proof Step . Multiplying (.) by u η , with any q > . Integrating over and noticing that function such that ∂u ∂t u(·, t)|∂ = . mq +  dt ≤ K – Since |∇umη|p(x) ≥ |∇umη|p – , we deal with the second term on the left-hand side of (.) as follows. ≥ q ≥ q = q mq +  dt ≤ K dx – q p– + q –  dx – q dx + q p– + q –  dx + q Combining (.) and (.), we have and Young’s inequality with to deduce dx + q ≤ K ≤ C ≤  dx + C( )C mp––m– m(p–+q–) dx + C( )C p– . mq+(x, t) dx ≤ C, u η p– + q –  ≤ K dx dt + q Choosing  and  appropriately, we have from (.) and (.) – for any q > , where C depends on q, p , m, and . dx dt ≤ dx dt ≤ C. considering (.), we obtain Similarly to (.), we obtain dx dt ≤ C, mq+(x, t) dx dt ≤ C, u η mq+(x, t) dx ≤ C. u η From (.) and (.), we conclude mq+(x, t) dx ≤ C + C(t – t), u η mq+(x, τ ) dx = u η mq+(x, τ + T ) dx ≤ C + CT . u η We finally arrive at for any q > , where C depends on q, p–, m, T and . Step . Let Ak(t) = x ∈ ; u η(x, t) > k , Ak(t) , ∇ umη + u η  + η ≤ K ∇ umη + u η  + η m(u η – k)+m–∇(u η – k)+ p(x) dx ∇(u η – k)+m p(x) dx ∇(u η – k)m p– dx – Ak( ) . ≤ K ∇ umη + u η  + η p(x)–  ∇ umη + u η ∇(u η – k)+m dx dt ∇ umη + u η  + η p(x)–  ∇ umη + u η ∇(u η – k)+m dx dt Substituting (.) into (.), we have dx ≤ K ≤ S p–(N+p–) if p– < N , where S is the Sobolev embedding constant, and r = ≤ K ≤ C Nr–N–p– ≤ C Ak( ) r(N+p–) Let Jk( ) = Nr–N–p– Utilizing Young’s inequality with , we obtain from (.) Nr–N–p– Nr–N–p– ≤ C Jk + C( )μk(p––)(N+p–) + Cμkp– . Nr–N–p– Upon choosing appropriately, one obtains Jk( ) ≤ C μ (p––)(N+p–) + μ p– k k Nr–N–p– For any h > k > , it is easy to see Jk( ) ≥ Ah( ) (h – k)rm. The relationships (.) and (.) above imply that deg u – T η , u+ , BR,  = , where u+ = max{u, }. Proposition . Assume that a ∈ L∞(QT ), K ∈ L∞(QT ). If u η solves u = G η(σ , ρf (u+) +  – σ ), for some σ ∈ [, ] and ρ ∈ [, ], then u η ≥  for any (x, t) ∈ QT . Moreover, if u η = , then u η >  in QT . ⎧⎨ – u = μu, x ∈ , ⎩ u = , x ∈ ∂ , γp+ = max γp(x) = p–p––  , M = K L(QT ) + p+– γp– = min γp(x) = p+p–+  , Cp+,p(x) max |∇e|| | p+ × max  γp±– a L(QT ) + | |T γp± T γp± = a – ∇ σ umη + u η  + η ∇ σ umη + u η  + η = – := (R). (R) = QT ∇ σ umη + u η  + η ∇ σ umη + u η  + η ∇ σ umη + u η  + η (R) ≤ ≤   =   × max  Since r < r ≤ ( m ) m– and  < <  , we have σ mumη– +  ≤ mumη– + <  + < . Hence   Noting that γ ± <  and using Hölder’s inequality, we have p ≤ T Integrating (.) over [, T ] and noting (.), we get × max QT QT ∇ σ umη + u η p(x) u η Lm∞+(QT ) + u η L∞(QT ) a L(QT ) ≤ rm+ + r a L(QT ) + rm + r | |T ≤ r a L(QT ) + | |T . Substituting this inequality into (.), we have × max r a L(QT ) + | |T × max p+– γ– Substituting (.) into (.) and noticing that    p ≤ p+–, we get (R) ≤ p+– × max  a L(QT ) + | |T γp± T γp± × max |∇φ| dx dt + p+– Step . We claim ≤ p+– × max ≤ p+– × max × max  γp±– a L(QT ) + | |T γp± T γp± e dx dt  γp±– a L(QT ) + | |T γp± T γp± Cp+,p(x) max |∇e|| | p+  γp±– a L(QT ) + | |T γp± T γp± Now the definition of r and (.) yield r ≤ ≤ r, 4 Existence of nontrivial nonnegative solution to (1.1) Theorem . Assume K (x, t) ≥  for a.e. (x, t) ∈ QT and T QT ae dx dt > μ. Then problem (.) has a nontrivial nonnegative periodic solution. deg u – G η , f u+ , BR\Br,  =  ∇ umη + u η  + η Multiplying (.) by umη + u η, integrating over QT and noting the T -periodicity of u η and the boundedness of u η, we have ≤ C ∇ umη + u η  + η ∇ umη + u η  dx dt umη+ + uη dx dt ≤ M, [,T]∩{t:|∇umη|Lp(x)( )>} [,T]∩{t:|∇umη|Lp(x)( )≤} [,T]∩{t:|∇umη|Lp(x)( )>} ∇umη p(x) dx dt + T ∇umη p(x) dx dt + T ≤ M + T . ∇ umη + u η  + η ∇ umη + u η  dx dt ∇ umη + u η p(x) dx dt. From (.) and (.), we have ∇ umη + u η p(x) dx dt ≤ C. ≤ C,p(x) max ∇ umη + u η p(x) dx So umη ∈ Lp– (, T ; W,p(x)( )) and umη is uniformly bounded in the space Lp– (, T ; W,p(x)( )). Thus, up to subsequence if necessary, we may assume that umη um ∈ Lp– (, T ; W,p(x)( )). In what follows, our main goal is to prove that u is a weak solution of problem (.). Step . The following relation is obvious:  ∇ umη + u η p(x) dx dt p± T – p± . QT QT QT ∇ umη + u η L( ) dt ≤ C,p(x) max ∇ umη + u η  dx dt ≤ C. In the following, we prove First, denote ∇ umη + u η  + η p(x)– p(x)  ∇ umη + u η p(x)– dx dt ≤ C. K–p = min Kp(x), K+p = max Kp(x), K–p = min Kp(x), K+p = max Kp(x), K–p = min Kp(x), K+p = max Kp(x). A straightforward computation shows that ≤  p+– K+p ≤  p+– K+p K+p ≤ Mp QT By the p(x)-Hölder’s inequality, we have p(x)– dx dt p(x)– dx dt m m ∇ u η + u η + ∇ u η + u η p(x)– dx dt p(x)– dx dt dx dt + p(x)– dx dt . × max p(x)– dx dt × max ∂t H ∈ (L p(x)– (QT ))N such that – in Lp (, T ; Lp(x)( )). Thus, we have Step . Using a method analogous to [], we get – ,p(x) of and η. Since umη is uniformly bounded in Lp (, T ; W  = ∂t + H∇ϕ – auϕ + uϕ ∇um p(x)–∇um∇ϕ dx dt = H∇ϕ dx dt. – (∇v), ∇ umη + u η – ∇v which gives  ≤ ∇ umη + u η  + η p(x)–  ∇v ∇ umη + u η – v dx dt p(x)– ∇ umη + u η  + η ∇ umη + u η  dx dt ∇ umη + u η  + η Multiplying the equation = a – ∇ umη + u η  + η ∇ umη + u η  + η ∇ umη + u η  dx dt ∇ umη + u η ∇v dx dt ∇v∇ H∇um dx dt = H∇v dx dt + |∇v|p(x)–∇v∇ um – v dx dt QT QT Thus, (.) and (.) imply  ≤  ≤  ≤  ≥ QT Combining (.) with (.), we obtain ∇ϕ dx dt. H – ∇u m p(x)– ∇u m ∇ϕ dx dt. H – ∇u m p(x)– ∇u m ∇ϕ dx dt. Proof of Lemma . Define the following sequence: ks = k + d – , s = , , , . . . , s = , , , . . . . ϕ(ks+) ≤ By induction, one has ϕ(ks) ≤ , s = , , , . . . , ϕ(ks+) ≤ where r >  is to be chosen. In fact, if (.) is right, then duced the result. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and approved the final version. Acknowledgements The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by National Science Foundation of China (11271154), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University. 1. Ru˚žicˇka , M: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748 . Springer, Berlin ( 2000 ) 2. DiBenedetto , E: Degenerate Parabolic Equations . Universitext. Springer, New York ( 1993 ) 3. Fragnelli , G, Nistri , P, Papini, D: Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms . Discrete Contin. Dyn. Syst . 31 , 35 - 64 ( 2011 ) 4. 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Zhongqing Li, Wenjie Gao. Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent, Boundary Value Problems, 2014, 77,