Abstract
We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are provided to illustrate our obtained results. In the proofs of our main results, we make use of the test function method and some integral inequalities.
Highlights
The Burgers equation∂t u + u∂ x u = ν∂ xx u, Fractional-in-Space Burgers-TypeEquations
Where ν > 0 is a certain parameter, is a fundamental partial differential equation arising in many physical problems, such as fluid mechanics, nonlinear acoustics, gas dynamics and traffic flow
In [2,3], Burgers used this equation to capture some features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and diffusion
Summary
2021, 5, 249 where 0 < α < 1 and ∂0α|t are the time-Caputo fractional derivative of order α, Kirane et al established a maximum principle, when the initial value u(0, ·) is sufficiently smooth. They discussed the influence of gradient nonlinearity on the global solvability of (3). We discuss the finite-time blow-up for the fractional-in-space analogue of (2) on a bounded interval, namely,.
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