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We study nonlinear pantograph-type reaction-diffusion PDEs, which, inaddition to the unknown $u=u(x,t)$, also contain the same functions withdilated or contracted arguments of the form $w=u(px,t)$, $w=u(x,qt)$, and$w=u(px,qt)$, where $p$ and $q$ are the free scaling parameters (for equationswith proportional delay we have $0<p<1$, $0<q<1$). A brief review ofpublications on pantograph-type ODEs and PDEs and their applications is given.Exact solutions and reductions of various types of such nonlinear partialfunctional differential equations are described for the first time. We presentexamples of nonlinear pantograph-type PDEs with proportional delay, which admittraveling-wave and self-similar solutions (note that PDEs with constant delaydo not have self-similar solutions). Additive, multiplicative and functionalseparable solutions, as well as some other exact solutions are also obtained.Special attention is paid to nonlinear pantograph-type PDEs of a rather generalform, which contain one or two arbitrary functions. In total, more than fortynonlinear pantograph-type reaction-diffusion PDEs with dilated or contractedarguments, admitting exact solutions, have been considered. Multi-pantographnonlinear PDEs are also discussed. The principle of analogy is formulated,which makes it possible to efficiently construct exact solutions of nonlinearpantograph-type PDEs. A number of exact solutions of more complex nonlinearfunctional differential equations with varying delay, which arbitrarily dependson time or spatial coordinate, are also described. The presented equations andtheir exact solutions can be used to formulate test problems designed toevaluate the accuracy of numerical and approximate analytical methods forsolving the corresponding nonlinear initial-boundary value problems for PDEswith varying delay.

Exact Solutions and Reductions of Nonlinear Diffusion PDEs of Pantograph Type