Liouville theorems for superlinear parabolic problems

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1 Liouville theorems for superlinear parabolic problems Pavol Quittner Comenius University, Bratislava Workshop in Nonlinear PDEs Brussels, September 7-11, 2015

2 Tintin & Prof. Calculus (Tournesol) c Hergé

3 Organizers: D. Bonheure (chair), J. B. Casteras, J. Főldes, B. Noris, M. Nys, A. Saldaña, C. Troestler Thank you!

4 Introduction Liouville-type theorems for entire solutions (t (, )) of scaling invariant superlinear parabolic problems guarantee optimal universal estimates for solutions of more general problems including estimates of their singularities and decay. We first consider several problems with gradient structure and show that each positive bounded entire solution has to be time-independent. Then we consider a class of two-component systems without gradient structure and show that the components of any positive bounded entire solution have to be proportional.

5 Liouville theorems for the model problem u t u = u p, x R n, t R (1) We will always assume p > 1 and u 0. Existence of stationary solutions: Gidas, Spruck 1981; Chen, Li 1991 (1) possesses positive stationary solutions iff n > 2 and p n+2. Sufficient conditions for nonexistence of positive solutions of (1) (i) p n+2 n... [Fujita 1966; Hayakawa 1973; Kobayashi, Sirao, Tanaka 1977] (ii) p < n(n+2) (n 1) 2... [Bidaut-Véron 1998] (iii) p < n+2 if u = u( x,t)... [Poláčik, Q., Souplet ] (iv) p < n... [Q ]

6 Liouville theorems for the model problem u t u = u p, x R n, t R (1) We will always assume p > 1 and u 0. Existence of stationary solutions: Gidas, Spruck 1981; Chen, Li 1991 (1) possesses positive stationary solutions iff n > 2 and p n+2. Sufficient conditions for nonexistence of positive solutions of (1) (i) p n+2 n... [Fujita 1966; Hayakawa 1973; Kobayashi, Sirao, Tanaka 1977] (ii) p < n(n+2) (n 1) 2... [Bidaut-Véron 1998] (iii) p < n+2 if u = u( x,t)... [Poláčik, Q., Souplet ] (iv) p < n... [Q ]

7 Liouville theorems for the model problem u t u = u p, x R n, t R (1) We will always assume p > 1 and u 0. Existence of stationary solutions: Gidas, Spruck 1981; Chen, Li 1991 (1) possesses positive stationary solutions iff n > 2 and p n+2. Sufficient conditions for nonexistence of positive solutions of (1) (i) p n+2 n... [Fujita 1966; Hayakawa 1973; Kobayashi, Sirao, Tanaka 1977] (ii) p < n(n+2) (n 1) 2... [Bidaut-Véron 1998] (iii) p < n+2 if u = u( x,t)... [Poláčik, Q., Souplet ] (iv) p < n... [Q ]

8 FROM ENTIRE SOLUTIONS TO STEADY STATES

9 u t u = u p in R n R...(1) 1 < p < n u 0 Idea of the proof: Assume that u 0 is a solution of (1). Doubling and scaling arguments we can assume u 1. Set E(ϕ) := R n ( 1 2 ϕ 2 1 p +1 ϕp+1) dx. Formally: d dt E(u(,t)) = u t (x,t) 2 dx R n u should be either an equilibrium or a heteroclinic orbit between (two sets of) equilibria; elliptic Liouville u 0. But: E(u(,t)) need not be defined, [Fila, Yanagida 2011]: p > n+2 homoclinic orbits: lim t ± u(,t) = 0, u > 0 is bounded, radially symmetric and spatially decaying if p < n 4 n 10

10 u t u = u p in R n R...(1) 1 < p < n u 0 For k = 1,2,... set w k (y,s) := (k t) β u(y k t,t), s = log(k t), t < k, where β = 1 p 1. Then w = w k solve the problem w s = 1 ρ (ρ w) βw +wp in R n R, ρ(y) = e y 2 /4, (1 ) E(ϕ) := R n ( 1 2 ϕ 2 + β 2 ϕ2 1 p+1 ϕp+1) ρdx is well defined for ϕ = w(,s) and d ds E(w(,s)) = R n w s (y,s) 2 dy. R n [(1 ) ρ] R n w(y,s)ρ(y)dy + s s 1 R n w p (y,s)ρ(y)dy ds C Set s k := logk. Then sk w k s k 1 R n s (y,s) 2 ρ(y)dy ds = E(wk (,s k 1)) E(w k (,s k )) (E) C sup s k 2 s s k 1 w k (,s) C(n,p)k β

11 u t u = u p in R n R...(1) 1 < p < n u 0 For k = 1,2,... set w k (y,s) := (k t) β u(y k t,t), s = log(k t), t < k, where β = 1 p 1. Then w = w k solve the problem w s = 1 ρ (ρ w) βw +wp in R n R, ρ(y) = e y 2 /4, (1 ) E(ϕ) := R n ( 1 2 ϕ 2 + β 2 ϕ2 1 p+1 ϕp+1) ρdx is well defined for ϕ = w(,s) and d ds E(w(,s)) = R n w s (y,s) 2 dy. R n [(1 ) ρ] R n w(y,s)ρ(y)dy + s s 1 R n w p (y,s)ρ(y)dy ds C Set s k := logk. Then sk w k s k 1 R n s (y,s) 2 ρ(y)dy ds = E(wk (,s k 1)) E(w k (,s k )) (E) C sup s k 2 s s k 1 w k (,s) C(n,p)k β

12 u t u = u p in R n R...(1) Summary: 1 < p < n u 0 Set w k (y,s) := (k t) β u(y k t,t), s = log(k t), t < k, sk w k s (y,s) 2 ρ(y)dy ds C(n,p)k β, s k := logk. (E) s k 1 R n v k (z,τ) := λ 2/(p 1) k w k (λ k z,λ 2 k τ+s k), z R n, k τ 0, λ k := 1 k. Then 0 k z < k v k (z,τ) = e βτ/k u ( e τ/2k z,k(1 e τ/k ) ) u(z,τ), v k 2 dz dτ = λ n+2+4/(p 1) sk τ k w k 2 dy ds 0 s s k 1 y <1 due to p < n and (E), hence u t 0. Elliptic Liouville u 0.

13 u t u = u p in R n R...(1) Summary: 1 < p < n u 0 Set w k (y,s) := (k t) β u(y k t,t), s = log(k t), t < k, sk w k s (y,s) 2 ρ(y)dy ds C(n,p)k β, s k := logk. (E) s k 1 R n v k (z,τ) := λ 2/(p 1) k w k (λ k z,λ 2 k τ+s k), z R n, k τ 0, λ k := 1 k. Then 0 k z < k v k (z,τ) = e βτ/k u ( e τ/2k z,k(1 e τ/k ) ) u(z,τ), v k 2 dz dτ = λ n+2+4/(p 1) sk τ k w k 2 dy ds 0 s s k 1 y <1 due to p < n and (E), hence u t 0. Elliptic Liouville u 0.

14 Vector valued generalization of (1) U = (u 1,u 2,...,u m ) 0 where U t U = F(U) in R n R, (2) F = G, G C 2+α loc (R m ), G(U) > G(0) for U 0, F(λU) = λ p F(U) for U 0, λ > 0, ξ F(U) > 0 for some ξ (0, ) m and all U > 0. Sufficient condition for nonexistence of positive solutions of (2) p < n (or p < n+2 if U(, t) is radially symmetric) Special case of (2): U = (u,v) 0, p = 2r +1, λ < 1 } u t u = u p λu r v r+1 v t v = v p λu r+1 v r in R n R (2 ) Known sufficient conditions (in the non-radial case): Fujita-type results: p n+2 n Bidaut-Véron s approach: n = 1, λ < r 3r+2... [Phan 2015] p < n(n+2), λ 0... (n 1) 2 [Phan, Souplet: preprint]

15 Vector valued generalization of (1) U = (u 1,u 2,...,u m ) 0 where U t U = F(U) in R n R, (2) F = G, G C 2+α loc (R m ), G(U) > G(0) for U 0, F(λU) = λ p F(U) for U 0, λ > 0, ξ F(U) > 0 for some ξ (0, ) m and all U > 0. Sufficient condition for nonexistence of positive solutions of (2) p < n (or p < n+2 if U(, t) is radially symmetric) Idea in the radial case: Let U be a positive radial solution of (2). 1. Scaling, doubling and Liouville for n = 1 decay estimates (as x ) U(,t) belongs to the energy space 2. Lyapunov functional U is a connecting orbit between equilibria 3. Elliptic Liouville no positive equilibria contradiction

16 Vector valued generalization of (1) U = (u 1,u 2,...,u m ) 0 where U t U = F(U) in R n R, (2) F = G, G C 2+α loc (R m ), G(U) > G(0) for U 0, F(λU) = λ p F(U) for U 0, λ > 0, ξ F(U) > 0 for some ξ (0, ) m and all U > 0. Sufficient condition for nonexistence of positive solutions of (2) p < n (or p < n+2 if U(, t) is radially symmetric) Special case of (2): U = (u,v) 0, p = 2r +1, λ < 1 } u t u = u p λu r v r+1 v t v = v p λu r+1 v r in R n R (2 ) Known sufficient conditions (in the non-radial case): Fujita-type results: p n+2 n Bidaut-Véron s approach: n = 1, λ < r 3r+2... [Phan 2015] p < n(n+2), λ 0... (n 1) 2 [Phan, Souplet: preprint]

17 Nonlinear boundary conditions u t u = 0 u ν = u q in R n + R, on R n + R, } (3) R n + := {(x R n : x 1 > 0}, x = (x 1,x 2,...,x }{{ n ) } =: x ν = ( 1,0,0,...,0), q > 1, u 0 Suff. conditions for nonexistence of bounded positive solutions of (3) q < n 1 (or q < n if u is axially symmetric: u = u(x 1, x,t)) Known results: Fujita-type: q n+1 n... [Galaktionov, Levine 1996], [Deng, Fila, Levine 1994] Results for solutions with bounded derivatives if n = 1... [Q., Souplet 2011] Condition q < n is optimal for the nonexistence of stationary solutions... [Hu 1994].

18 Nonlinear boundary conditions u t u = 0 u ν = u q in R n + R, on R n + R, } (3) R n + := {(x R n : x 1 > 0}, x = (x 1,x 2,...,x }{{ n ) } =: x ν = ( 1,0,0,...,0), q > 1, u 0 Suff. conditions for nonexistence of bounded positive solutions of (3) q < n 1 (or q < n if u is axially symmetric: u = u(x 1, x,t)) Known results: Fujita-type: q n+1 n... [Galaktionov, Levine 1996], [Deng, Fila, Levine 1994] Results for solutions with bounded derivatives if n = 1... [Q., Souplet 2011] Condition q < n is optimal for the nonexistence of stationary solutions... [Hu 1994].

19 FROM SYSTEMS TO SCALAR EQUATIONS

20 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) v t v = v r ( b 2 v q +c 2 u q ) in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) Sufficient conditions for nonexistence of positive solutions of (4) q +r < max ( n(n+2) ) n, (n 1) 2 (or q +r < n+2 if u,v are radially symmetric) q = r = 1 (Lotka-Volterra): sufficient (and necessary) condition n 5 q = 2, r = 1: sufficient (and necessary) condition n 3 u t u = b 1 u 3 +c 1 uv 2 v t v = b 2 v 3 +c 2 u 2 v

21 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) =: f v t v = v r ( b 2 v q +c 2 u q ) =: g in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) [Montaru, Souplet, Sirakov 2014] K > 0 (f Kg)(u }{{}} Kv {{} ) 0 w t w =:w Aim: Show u = Kv (i.e. w = 0), then u t u = cu q+r for some c > 0. Sufficient conditions for nonexistence of positive solutions of (4) q +r < max ( n(n+2) ) n, (n 1) 2 (or q +r < n+2 if u,v are radially symmetric) q = r = 1 (Lotka-Volterra): sufficient (and necessary) condition n 5 q = 2, r = 1: sufficient (and necessary) condition n 3 u t u = b 1 u 3 +c 1 uv 2 v t v = b 2 v 3 +c 2 u 2 v

22 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) =: f v t v = v r ( b 2 v q +c 2 u q ) =: g in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) [Montaru, Souplet, Sirakov 2014] K > 0 (f Kg)(u }{{}} Kv {{} ) 0 w t w =:w Aim: Show u = Kv (i.e. w = 0), then u t u = cu q+r for some c > 0. Idea of the proof of w = 0 (for u,v bounded): (w t w)sign(w) h( w ) in R n R, where h C([0, )), h(s) > 0 for s > 0. Assume on the contrary w 0. W.l.o.g. w(x,t ) > 0 for some x,t. Then we arrive at a contradiction by considering the points of maxima of cf. [Főldes 2011]. w ε (x,t) := w(x,t) ε x x 2 ε ( (t t ) ),

23 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) v t v = v r ( b 2 v q +c 2 u q ) in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) Sufficient conditions for nonexistence of positive solutions of (4) q +r < max ( n(n+2) ) n, (n 1) 2 (or q +r < n+2 if u,v are radially symmetric) q = r = 1 (Lotka-Volterra): sufficient (and necessary) condition n 5 q = 2, r = 1: sufficient (and necessary) condition n 3 u t u = b 1 u 3 +c 1 uv 2 v t v = b 2 v 3 +c 2 u 2 v

24 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) v t v = v r ( b 2 v q +c 2 u q ) in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) Sufficient conditions for nonexistence of positive solutions of (4) q +r < max ( n(n+2) ) n, (n 1) 2 (or q +r < n+2 if u,v are radially symmetric) q = r = 1 (Lotka-Volterra): sufficient (and necessary) condition n 5 q = 2, r = 1: sufficient (and necessary) condition n 3 u t u = b 1 u 3 +c 1 uv 2 v t v = b 2 v 3 +c 2 u 2 v

25 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) v t v = v r ( b 2 v q +c 2 u q ) in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) Sufficient conditions for nonexistence of positive solutions of (4) q +r < max ( n(n+2) ) n, (n 1) 2 (or q +r < n+2 if u,v are radially symmetric) q = r = 1 (Lotka-Volterra): sufficient (and necessary) condition n 5 Application: Existence of periodic solutions } of u t u = u(a 1 b 1 u +c 1 v) in Ω [0,T], v t v = v(a 2 b 2 v +c 2 u) (4 p ) u = v = 0 on Ω [0,T], where Ω R n is smooth and bounded, a i,b i,c i C(Ω [0,T]) are T-periodic in t and satisfy (5), a 1,a 2 < λ 1. Theorem. If n 5 then (4 p ) possesses a positive T-periodic solution.

26 Systems without gradient structure q r > 0, q +r > 1 u t u = u r ( b 1 u q +c 1 v q ) v t v = v r ( b 2 v q +c 2 u q ) in R n R, (4) b 1,b 2,c 1,c 2 > 0, c 1 c 2 > b 1 b 2 (5) Sufficient conditions for nonexistence of positive solutions of (4) q +r < max ( n(n+2) ) n, (n 1) 2 (or q +r < n+2 if u,v are radially symmetric) q = r = 1 (Lotka-Volterra): sufficient (and necessary) condition n 5 q = 2, r = 1: sufficient (and necessary) condition n 3 u t u = b 1 u 3 +c 1 uv 2 v t v = b 2 v 3 +c 2 u 2 v

27 Thank you for your attention!

Homoclinic and Heteroclinic Connections. for a Semilinear Parabolic Equation. Marek Fila (Comenius University)

Homoclinic and Heteroclinic Connections for a Semilinear Parabolic Equation Marek Fila (Comenius University) with Eiji Yanagida (Tohoku University) Consider the Fujita equation (E) u t =Δu + u p 1 u in