Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

Figure 2.1.  Illustration of the topology applied to problem (1.1) for scalar $u$. Model (1.1) exhibits steady states with jump discontinuity and global existence of classical solutions. $\tilde{u}$ represents a steady state while $u(t,x)$ represents a solution for some $t$

Figure 3.1.  Plot of the nullclines of $f_r(u,v)=-(1+v)u+m_1(u^2/(1+ku^2))$ for $u\neq 0$ and $g_r(u,v)=-(\mu_3+u)v+m_2 (u^2/(1+ku^2))$.

Figure 3.2.  Numerically obtained solution to model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=1$. We observe convergence towards a steady state with jump discontinuity. Left: Non-diffusive component $u$. Right: Diffusive component $v$.

Figure 3.3.  Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01$ with varying diffusion coefficient: upper left: $D=5$, upper right: $D=1$, lower left: $D=0.5$, lower right: $D=0.1$. We observe emergence of more jump-type discontinuities for smaller diffusion coefficient. Initial conditions are $u_0(x)=1.725-0.1\cos(2 \pi x^2), v_0(x)=2.48615$

Figure 3.4.  Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=5$. We observe emergence of jump-type discontinuities around local maxima of the initial conditions. Initial conditions are $u_0(x)=1.725-0.1x^4\cos(8 \pi x^2), v_0(x)=2.48615$

Figure 5.1.  Nullclines of $f_r$ and $g_r$ and $v_{f_r,g_r}$ for parameters $m_1 = 1.44, m_2 = 2, \mu_3 = 4.2, k=0.1$

Figure 5.2.  Illustration of the right-hand side of $-\partial^2 v/\partial x^2 = g_r(u,v)$ for different branches of the solution $u(v)$ of $u_t=f_r(u,v)=0$. The parameters for illustration are $D=1,m_1=1.44,m_2=2,\mu_3=4.2$. We can observe that all nontrivial homogeneous steady states are of type $u_-$

Figure 5.3.  Illustration of the construction of weak steady states in the proof of Lemma 3.6