On one-dimensional forward-backward diffusion equations with linear convection and reaction
In the present work, for arbitrary \(T>0\), the authors study an initial-Neumann boundary value problem for a class of one-dimensional forward-backward diffusion equations with linear convection and reaction of the form: \[ u_t=(\sigma (u_x))_x+b(x,t)u_x+c(x,t)u+f(x,t) \] in \(\Omega \times (0,T)\), \(\Omega =(0,1)\), with the following initial and Neumann boundary conditions \(u=u_0\) on \(\Omega \times\{t=0\}\) and \(u_x=0\) on \(\partial \Omega \times (0,T)\). It is assumed that the initial function \(u_0(x)\in C^{2+\alpha}(\bar{\Omega})\) for some \(\alpha \in (0,1)\) and satisfy the compatibility condition \(u'_0=0\) on \(\partial \Omega\). The diffusion flux function \(\sigma (s)\), \(s\in \mathbb R\) is assumed to contain two forward-diffusion phases, or more specifically, it is supposed that there exist numbers \(s_2>s_1>0\) and \(\Lambda \geq \lambda >0\) such that \(\sigma (0)=0\), \(\sigma (s_1)>\sigma (s_2)\geq 0\), \(\sigma \in C^{1+\alpha}(J_1)\), \(\sigma'\in L_{\mathrm{loc}}^1(J_1,(0,\infty ))\bigcap L_{\mathrm{loc}}^1(J_2,[\lambda ,\Lambda ])\) where \(J_k=\{(-\infty ,\frac{s_2}{k})\cup(\frac{s_1}{k},\infty )\}\), \(k=1,2\). It is proved that for all smooth initial functions with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. In addition, the authors introduce the notion of transition gauge for this kind of solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant.
scientific article; zbMATH DE number 6987671
scientific article