Systems of parabolic equations with uniformly continuous coefficients. (English) Zbl 0144.35203

Fabes, E. B.; Riviere, N. M.

J. Anal. Math. 17, 305-335 (1966).

The authors consider systems of partial differential operators of parabolic type, namely, \[ \sum_{j=1}^N \sum_{\vert\alpha\vert\le m} a_x^{ij}(x,t) \left( \frac{\partial}{\partial x}\right)^\alpha u_j(x, t) - \frac{\partial u_i(x,t)}{\partial t} = f_i(x, t), \] where \(i = 1,\ldots,n\), \(x\in E^n\) and \(t\in [0, R]\), \(0 < R <\infty\). The components \(f_i\) of \(f = (f_1,\ldots, f_N)\) belong to \(L^p(E^n\times (0, R))\), \(1 < p <\infty\). The authors seek solutions \(u(x, t) = (u_1,\ldots, u_N)\) with components \(u_i(x, t)\) belonging to \(L_0^{p,m,1}(E^n\times (0, R)) = \) closure of all \(\varphi(x, t)\in C_0^\infty(E\times (0, \infty))\) with respect to the norm \[ \Vert \varphi\Vert = \sum_{\vert\alpha\vert\le m} \left( \int_0^R \int_{E^n} \left\vert\left(\frac{\partial}{\partial x}\right)^\alpha \varphi \right\vert^p \,dx\,dt\right)^{1/p} + \left( \int_0^R \int_{E^n} \left\vert \frac{\partial \varphi}{\partial t} \right\vert^p \,dx\,dt\right)^{1/p}, \] admitting that \(u(x, t)\) in this generalized sense satisfy the condition \(u(x, 0) = 0\). All the coefficients \(a_x^{ij}\) are assumed to be bounded and the coefficients of the highest order are uniformly continuous. Basing on the properties of singular integral operators \[ \int_0^{t-\varepsilon} \int_{E^n} K(x, t; x - y, t - s) f(y, s)\,dy\,ds, \quad \varepsilon > 0, \] defined in Part 1 of this paper and using the ideal of pseudo-product, pseudo-inverse and symbol, familiar to the singular integrals of A. P. Calderón and A. Zygmund [Am. J. Math. 79, 901–921 (1957; Zbl 0081.33502)] the authors prove existence and uniqueness of the solution in this generalized sense.

Reviewer: J. Wolska-Bochenek

MSC:

35K41

Higher-order parabolic systems

Keywords:

systems of parabolic equations; uniformly continuous coefficients

Citations:

Zbl 0081.33502

References:

[1]	Calderón, A. P.; Zygmund, A., Singular integral operators and differential equations, Am. J. Math., 79, 901-921 (1957) · Zbl 0081.33502 · doi:10.2307/2372441
[2]	Fabes, E., Singular integrals and partial differential equations of parabolic type, to appear inStudia Math. · Zbl 0144.35002
[3]	Jones, B. F., A class of singular integrals, Am. J. Math., 86, 441-462 (1964) · Zbl 0123.08501 · doi:10.2307/2373175