Instantaneous blow-up for a Sobolev-type equation arising in the theory of propagation of nonlinear waves in semiconductors

This work is concerned with the study of the instantaneous blow-up phenomenon for the Sobolev-type equation $ \frac{\partial^2u}{\partial t\partial x}+\frac{\partial }{\partial x}\left(|u|^{p-2}u\right) = |u|^q $ in $ (0, T)\times I $, where $ u = u(t, x) $, $ p\geq 2 $, and $ q>p-1 $. The considered equation arises in the theory of propagation of nonlinear waves in semiconductors. The problem is first studied on the half-line $ I = (0, \infty) $ under the inhomogeneous boundary condition $ u(t, 0) = \delta>0 $. Next, the fractional version of the above problem is studied, namely $ \frac{\partial}{\partial t}\frac{\partial^\alpha u} {\partial x^\alpha}+\frac{\partial^\alpha }{\partial x^\alpha}\left(|u|^{p-2}u\right) = |u|^q $ in $ (0, T)\times (0, \infty) $, where $ 0<\alpha<1 $ and $ \frac{\partial^\alpha} {\partial x^\alpha} $ is the Liouville-Caputo fractional derivative of order $ \alpha $ with respect to the space-variable $ x $. Finally, the problem is discussed on the segment $ I = [0, L] $ with a singular weight function of the form $ x^{-\sigma} $, $ \sigma>0 $, in front of the nonlinearity $ |u|^q $, under the inhomogeneous boundary condition $ u(t, L) = \delta<0 $. We point out that in all the considered problems, no restriction on the initial function $ u(0, \cdot) $ is imposed. In the first case, an interesting phenomenon is observed. Namely, the inhomogeneous term $ \delta $, no matter how small it is, has the effect of producing the instantaneous blow-up. In the second case, the same result holds. Namely, the value of the order of the fractional derivative $ \alpha $ has no effect on the instantaneous blow-up. In the third case, we show that $ \sigma^* = 1 $ is critical in the following sense: (ⅰ) If $ \sigma>\sigma^* $, then an instantaneous blow-up occurs; (ⅱ) If $ \sigma<\sigma^* $, then local weak solutions exist for some $ \delta<0 $.