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By Jeffrey A.

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We consider ut + uX= 0 in Il8 x (0, oo), onTR. It is easy to verify that {t = 0} is noncharacteristic. The characteristic ODE and corresponding initial values are given by dt dx ds 1' ds 1' and t(0) = 0. Here, both x and t are treated as functions of s. Hence x(0) = XO, x=s+xp, t=s. By eliminating s, we have x - t=xo. This is a straight line containing (xo, 0) and with a slope 1. Along this straight line, u is constant. Hence u(x,t) = uo(x - t). 1. With t as time, the graph of the solution represents a wave propagating to the right with velocity 1 without changing shape.

For a general firstorder nonlinear PDE, the corresponding ordinary differential system consists of 2n + 1 equations for 2n +1 functions x, u and Du. Here, we need to take into account the gradient of u by adding n more equations for Du. In other words, we regard our first-order nonlinear PDE as a relation for (u, p) with a constraint p = Du. We should emphasize that this is a unique feature for single first-order PDEs. For PDEs of higher order or for first-order partial differential systems, nonlinear equations are dramatically different from linear equations.

A natural question here is whether there exists a global solution for globally defined a and uo. There are several reasons that local solutions cannot be extended globally. First, u(x) cannot be evaluated at x E I[8n if x is not on an integral curve from the initial hypersurface, or equivalently, the integral curve from x does not intersect the initial hypersurface. Second, u(x) cannot be evaluated at x E Il8n if the integral curve starting from x intersects the initial hypersurface more than once.

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Applied partial differential equations. An introduction by Jeffrey A.


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Categories: Differential Equations