Existence of infinitely many solutions for a forward backward heat equation

Abstract

Let ϕ \phi be a piecewise linear function which satisfies the condition s ϕ ( s ) ⩾ c s 2 , c > 0 , s ∈ R s\phi (s) \geqslant c{s^2},c > 0,s \in {\mathbf {R}} , and which is monotone decreasing on an interval ( a , b ) ⊂ R + (a,b) \subset {{\mathbf {R}}_ + } . It is shown that for f ∈ C 2 [ 0 , 1 ] f \in {C^2}[0,1] , with max f ′ > a \max f^\prime > a , there exists a T > 0 T > 0 such that the initial boundary value problem \[ u t = ϕ ( u x ) x , u x ( 0 , t ) = u x ( 1 , t ) = 0 , u ( ⋅ , 0 ) = f , {u_t} = \phi \,{({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f, \] has infinitely many solutions u u satisfying ∥ u ∥ α , ∥ u x ∥ ∞ , ∥ u t ∥ 2 ⩽ c ( f , ϕ ) \parallel \;u\;{\parallel _{\alpha }},\parallel \;{u_x}{\parallel _{\infty }},\parallel \;{u_t}{\parallel _{2}} \leqslant c(f,\phi ) on [ 0 , 1 ] × [ 0 , T ] [0,1] \times [0,T] .