Existence of infinitely many solutions for a forward backward heat equation

Abstract:

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