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Boundary value problems for the diffusion equation with piecewise continuous time delay. (English) Zbl 0879.35065
The following partial differential equations with piecewise constant argument are considered: \[ u_t(x,t)=a^2u_{xx}(x,t)-bu(x,[t]), \qquad u_t(x,t)=a^2u_{xx}(x,t)-bu\Bigl(x,\bigl[t+{\textstyle{1\over2}}\bigr]\Bigr), \] \((x,t)\in[0,1]\times[0,\infty)\), subject to homogeneous boundary conditions \(u(0,t)=u(1,t)=0\) and initial condition \(u(x,0)=u_0(x)\). The solution is obtained by the usual separation of variables method. Using the obtained explicit expressions for the solutions, various qualitative properties such as boundedness, oscillation, and stability are obtained. The same study for the parabolic equation of neutral type \(u_t(x,t)=a^2u_{xx}(x,t)+bu_t(x,[t])\) is undertaken.
Reviewer: I.Ginchev (Varna)

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
35R10 Functional partial differential equations
35K15 Initial value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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