Weak solutions of hyperbolic-parabolic Volterra equations

Gripenberg, Gustaf

doi:10.1090/S0002-9947-1994-1216335-0

Weak solutions of hyperbolic-parabolic Volterra equations
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by Gustaf Gripenberg

Trans. Amer. Math. Soc. 343 (1994), 675-694

DOI: https://doi.org/10.1090/S0002-9947-1994-1216335-0

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Abstract:

The existence of a global weak solution, satisfying certain a priori ${L^\infty }$-bounds, of the equation ${u_t}(t,x) = \int _0^tk(t - s){(\sigma ({u_x}))_x}(s,x)ds + f(t,x)$ is established. The kernel k is locally integrable and log-convex, and $\sigma \prime$ has only one local minimum which is positive.

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