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Nonlinear singular integral inequalities for functions in two and $$n$$ independent variables. (English) Zbl 0985.26010
The author establishes a priori bounds on solutions of some nonlinear integral inequalities with weakly singular kernels for functions in two and $$n$$ independent variables. The typical inequality considered is of the form $u(x,y)\leq a(x,y)+ \int^x_0 \int^y_0 (x- s)^{a-1}(y- t)^{\beta- 1}F(s,t) \omega(u(s,t)) ds dt,\;x,y\in [0,T),\tag{$$*$$}$ where $$0< T\leq \infty$$, $$\alpha> 0$$, $$\beta> 0$$, $$u,F:[0,T)^2\to R'= [0,\infty)$$ are continuous functions, $$\omega: R^+\to R^+$$ is a $$C^1$$-function, and $$a(x,y): [0,T)^2\to R'$$ is a $$C^2$$-function satisfying the condition $$a_{xy}(x,y)\geq 0$$, $$a_x(x,y)\geq 0$$ [or $$a_y(x,y)\geq 0$$].
In Theorem 2.2 (the main result), a priori bounds on solutions of inequality $$(*)$$ are obtained for two cases: (i) $$\alpha>{1\over 2}$$, $$\beta>{1\over 2}$$ and $$\omega$$ satisfies the condition $$(q)$$ with $$q= 2$$; and (ii) $$\alpha= \beta= 1/(z+ 1)$$ and $$\omega$$ satisfies the condition (q) with $$q= z+ 2$$, where $$z\geq 1$$ is a real number.
Definition. Let $$q> 0$$ be a real number. A function $$\omega: R^+\to R^+$$ is called satisfying a condition (q) if $e^{-qt}[\omega(u)]^q\leq R(t) \omega(e^{-qt} u^q),\quad u\in R^+,\quad 0\leq t< T,$ holds for some continuous function $$R(t): R^+\to R^+$$.
The argument used herein is an approach proposed by the present author in [J. Math. Anal. Appl. 214, No. 2, 349-366 (1997; Zbl 0893.26006)]. A singular integral inequality for $$n$$-independent-variable functions is also given.

##### MSC:
 26D10 Inequalities involving derivatives and differential and integral operators 35B35 Stability in context of PDEs 35K55 Nonlinear parabolic equations 45G05 Singular nonlinear integral equations
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